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## Statistics and probability

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Introductory Statistics: A Problem-Solving Approach

## Psychology in Everyday Life

Third edition | ©2020 stephen kokoska.

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## A Problem Solving Approach

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## 5 Teaching Mathematics Through Problem Solving

Janet Stramel

In his book “How to Solve It,” George Pólya (1945) said, “One of the most important tasks of the teacher is to help his students. This task is not quite easy; it demands time, practice, devotion, and sound principles. The student should acquire as much experience of independent work as possible. But if he is left alone with his problem without any help, he may make no progress at all. If the teacher helps too much, nothing is left to the student. The teacher should help, but not too much and not too little, so that the student shall have a reasonable share of the work.” (page 1)

What is a problem in mathematics? A problem is “any task or activity for which the students have no prescribed or memorized rules or methods, nor is there a perception by students that there is a specific ‘correct’ solution method” (Hiebert, et. al., 1997). Problem solving in mathematics is one of the most important topics to teach; learning to problem solve helps students develop a sense of solving real-life problems and apply mathematics to real world situations. It is also used for a deeper understanding of mathematical concepts. Learning “math facts” is not enough; students must also learn how to use these facts to develop their thinking skills.

According to NCTM (2010), the term “problem solving” refers to mathematical tasks that have the potential to provide intellectual challenges for enhancing students’ mathematical understanding and development. When you first hear “problem solving,” what do you think about? Story problems or word problems? Story problems may be limited to and not “problematic” enough. For example, you may ask students to find the area of a rectangle, given the length and width. This type of problem is an exercise in computation and can be completed mindlessly without understanding the concept of area. Worthwhile problems includes problems that are truly problematic and have the potential to provide contexts for students’ mathematical development.

There are three ways to solve problems: teaching for problem solving, teaching about problem solving, and teaching through problem solving.

Teaching for problem solving begins with learning a skill. For example, students are learning how to multiply a two-digit number by a one-digit number, and the story problems you select are multiplication problems. Be sure when you are teaching for problem solving, you select or develop tasks that can promote the development of mathematical understanding.

Teaching about problem solving begins with suggested strategies to solve a problem. For example, “draw a picture,” “make a table,” etc. You may see posters in teachers’ classrooms of the “Problem Solving Method” such as: 1) Read the problem, 2) Devise a plan, 3) Solve the problem, and 4) Check your work. There is little or no evidence that students’ problem-solving abilities are improved when teaching about problem solving. Students will see a word problem as a separate endeavor and focus on the steps to follow rather than the mathematics. In addition, students will tend to use trial and error instead of focusing on sense making.

Teaching through problem solving focuses students’ attention on ideas and sense making and develops mathematical practices. Teaching through problem solving also develops a student’s confidence and builds on their strengths. It allows for collaboration among students and engages students in their own learning.

Consider the following worthwhile-problem criteria developed by Lappan and Phillips (1998):

- The problem has important, useful mathematics embedded in it.
- The problem requires high-level thinking and problem solving.
- The problem contributes to the conceptual development of students.
- The problem creates an opportunity for the teacher to assess what his or her students are learning and where they are experiencing difficulty.
- The problem can be approached by students in multiple ways using different solution strategies.
- The problem has various solutions or allows different decisions or positions to be taken and defended.
- The problem encourages student engagement and discourse.
- The problem connects to other important mathematical ideas.
- The problem promotes the skillful use of mathematics.
- The problem provides an opportunity to practice important skills.

Of course, not every problem will include all of the above. Sometimes, you will choose a problem because your students need an opportunity to practice a certain skill.

Key features of a good mathematics problem includes:

- It must begin where the students are mathematically.
- The feature of the problem must be the mathematics that students are to learn.
- It must require justifications and explanations for both answers and methods of solving.

Problem solving is not a neat and orderly process. Think about needlework. On the front side, it is neat and perfect and pretty.

But look at the b ack.

It is messy and full of knots and loops. Problem solving in mathematics is also like this and we need to help our students be “messy” with problem solving; they need to go through those knots and loops and learn how to solve problems with the teacher’s guidance.

When you teach through problem solving , your students are focused on ideas and sense-making and they develop confidence in mathematics!

## Mathematics Tasks and Activities that Promote Teaching through Problem Solving

## Choosing the Right Task

Selecting activities and/or tasks is the most significant decision teachers make that will affect students’ learning. Consider the following questions:

- Teachers must do the activity first. What is problematic about the activity? What will you need to do BEFORE the activity and AFTER the activity? Additionally, think how your students would do the activity.
- What mathematical ideas will the activity develop? Are there connections to other related mathematics topics, or other content areas?
- Can the activity accomplish your learning objective/goals?

## Low Floor High Ceiling Tasks

By definition, a “ low floor/high ceiling task ” is a mathematical activity where everyone in the group can begin and then work on at their own level of engagement. Low Floor High Ceiling Tasks are activities that everyone can begin and work on based on their own level, and have many possibilities for students to do more challenging mathematics. One gauge of knowing whether an activity is a Low Floor High Ceiling Task is when the work on the problems becomes more important than the answer itself, and leads to rich mathematical discourse [Hover: ways of representing, thinking, talking, agreeing, and disagreeing; the way ideas are exchanged and what the ideas entail; and as being shaped by the tasks in which students engage as well as by the nature of the learning environment].

The strengths of using Low Floor High Ceiling Tasks:

- Allows students to show what they can do, not what they can’t.
- Provides differentiation to all students.
- Promotes a positive classroom environment.
- Advances a growth mindset in students
- Aligns with the Standards for Mathematical Practice

Examples of some Low Floor High Ceiling Tasks can be found at the following sites:

- YouCubed – under grades choose Low Floor High Ceiling
- NRICH Creating a Low Threshold High Ceiling Classroom
- Inside Mathematics Problems of the Month

## Math in 3-Acts

Math in 3-Acts was developed by Dan Meyer to spark an interest in and engage students in thought-provoking mathematical inquiry. Math in 3-Acts is a whole-group mathematics task consisting of three distinct parts:

Act One is about noticing and wondering. The teacher shares with students an image, video, or other situation that is engaging and perplexing. Students then generate questions about the situation.

In Act Two , the teacher offers some information for the students to use as they find the solutions to the problem.

Act Three is the “reveal.” Students share their thinking as well as their solutions.

“Math in 3 Acts” is a fun way to engage your students, there is a low entry point that gives students confidence, there are multiple paths to a solution, and it encourages students to work in groups to solve the problem. Some examples of Math in 3-Acts can be found at the following websites:

- Dan Meyer’s Three-Act Math Tasks
- Graham Fletcher3-Act Tasks ]
- Math in 3-Acts: Real World Math Problems to Make Math Contextual, Visual and Concrete

## Number Talks

Number talks are brief, 5-15 minute discussions that focus on student solutions for a mental math computation problem. Students share their different mental math processes aloud while the teacher records their thinking visually on a chart or board. In addition, students learn from each other’s strategies as they question, critique, or build on the strategies that are shared.. To use a “number talk,” you would include the following steps:

- The teacher presents a problem for students to solve mentally.
- Provide adequate “ wait time .”
- The teacher calls on a students and asks, “What were you thinking?” and “Explain your thinking.”
- For each student who volunteers to share their strategy, write their thinking on the board. Make sure to accurately record their thinking; do not correct their responses.
- Invite students to question each other about their strategies, compare and contrast the strategies, and ask for clarification about strategies that are confusing.

“Number Talks” can be used as an introduction, a warm up to a lesson, or an extension. Some examples of Number Talks can be found at the following websites:

- Inside Mathematics Number Talks
- Number Talks Build Numerical Reasoning

## Saying “This is Easy”

“This is easy.” Three little words that can have a big impact on students. What may be “easy” for one person, may be more “difficult” for someone else. And saying “this is easy” defeats the purpose of a growth mindset classroom, where students are comfortable making mistakes.

When the teacher says, “this is easy,” students may think,

- “Everyone else understands and I don’t. I can’t do this!”
- Students may just give up and surrender the mathematics to their classmates.
- Students may shut down.

Instead, you and your students could say the following:

- “I think I can do this.”
- “I have an idea I want to try.”
- “I’ve seen this kind of problem before.”

Tracy Zager wrote a short article, “This is easy”: The Little Phrase That Causes Big Problems” that can give you more information. Read Tracy Zager’s article here.

## Using “Worksheets”

Do you want your students to memorize concepts, or do you want them to understand and apply the mathematics for different situations?

What is a “worksheet” in mathematics? It is a paper and pencil assignment when no other materials are used. A worksheet does not allow your students to use hands-on materials/manipulatives [Hover: physical objects that are used as teaching tools to engage students in the hands-on learning of mathematics]; and worksheets are many times “naked number” with no context. And a worksheet should not be used to enhance a hands-on activity.

Students need time to explore and manipulate materials in order to learn the mathematics concept. Worksheets are just a test of rote memory. Students need to develop those higher-order thinking skills, and worksheets will not allow them to do that.

One productive belief from the NCTM publication, Principles to Action (2014), states, “Students at all grade levels can benefit from the use of physical and virtual manipulative materials to provide visual models of a range of mathematical ideas.”

You may need an “activity sheet,” a “graphic organizer,” etc. as you plan your mathematics activities/lessons, but be sure to include hands-on manipulatives. Using manipulatives can

- Provide your students a bridge between the concrete and abstract
- Serve as models that support students’ thinking
- Provide another representation
- Support student engagement
- Give students ownership of their own learning.

Adapted from “ The Top 5 Reasons for Using Manipulatives in the Classroom ”.

any task or activity for which the students have no prescribed or memorized rules or methods, nor is there a perception by students that there is a specific ‘correct’ solution method

should be intriguing and contain a level of challenge that invites speculation and hard work, and directs students to investigate important mathematical ideas and ways of thinking toward the learning

involves teaching a skill so that a student can later solve a story problem

when we teach students how to problem solve

teaching mathematics content through real contexts, problems, situations, and models

a mathematical activity where everyone in the group can begin and then work on at their own level of engagement

20 seconds to 2 minutes for students to make sense of questions

Mathematics Methods for Early Childhood by Janet Stramel is licensed under a Creative Commons Attribution 4.0 International License , except where otherwise noted.

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## AI Can Help You Ask Better Questions — and Solve Bigger Problems

- Hal Gregersen
- Nicola Morini Bianzino

Leaders should focus less on automation and more on innovation.

Most companies still view AI rather narrowly, as a tool that alleviates the costs and inefficiencies of repetitive human labor and increasing organizations’ capacity to produce, process, and analyze piles and piles of data. But when paired with “soft” inquiry-related skills it can help people ask better questions and be more innovative.

There are two distinct, yet related, paths to do this. 1) Use the technology to change the cadence and patterns of their questions: AI increases question velocity, question variety, and question novelty. 2) Use AI to transform the conditions and settings where people work so that questions that spark change — what we call “catalytic” questions — can emerge. This pushes leaders out of their comfort zones and into the position of being intellectually wrong, emotionally uncomfortable, and behaviorally quiet and more reflective, all of which, it turns out, promotes innovative thinking and action.

Just a few years ago, businesses wrestled with artificial intelligence mainly in the abstract — a “future of work” problem they’d have to contend with down the line. Now? More than half the companies around the world are actively adopting AI. Although investments are particularly high in industries such as health care, data management and processing, cloud computing, and fintech, all types of organizations and functions have incorporated AI technology into their operations. And generative AI tools such as ChatGPT are forcing leaders to ask where and how AI can help their businesses.

Even so, most companies still view AI rather narrowly, as a tool that alleviates the costs and inefficiencies of repetitive human labor by automating mundane physical tasks (like moving merchandise in warehouses) and increasing organizations’ capacity to produce, process, and analyze piles and piles of data. But the technology can do much more than that.

Paired with “soft” inquiry-related skills such as critical thinking, innovation, active learning, complex problem solving, creativity, originality, and initiative, this technology can further our understanding of an increasingly complex world, allowing us to engage in more abstract questioning and shifting our focus from identification to ideation. In our research and workshops with executives, we’re finding that companies have much to gain by treating AI as a knowledge-work collaborator in diverse areas such as product design, process efficiency, and prompt engineering. Partnering with the technology in this way can help people ask smarter questions , which in turn makes them better problem solvers and breakthrough innovators. We are also seeing the initial impacts of more context-aware AI systems (like ChatGPT), and as they continue to improve, the skill of asking questions (or creating prompts) will only become more valuable in the discovery process.

Although experts have recognized the need for software engineers to ask smart questions upstream, when developing automated tools (to bake in fewer biases and assumptions), little has been said about the flipside of the relationship between AI and inquiry: the technology’s potential to help people become more inquisitive, creative problem-solvers on the job. We aimed to correct this oversight through design-thinking sessions and extensive follow-up conversations with tech-driven business leaders from a diverse array of countries and industries. We also surveyed roughly 200 leaders, from more than 30 countries who participated in our executive education programs at MIT —to learn how artificial intelligence has affected questioning patterns and innovation behaviors and outcomes in their organizations. (For this research, we’ve defined “artificial intelligence” broadly to include machine learning, deep learning, robotics, and the recent explosion of generative AI.)

We have found two distinct, yet related, paths that leaders follow to strengthen their (and their teams’) inquiry muscles as they tap the power of AI in their question-asking work.

On the first path, they can use the technology to change the cadence and patterns of their questions: AI increases question velocity, question variety, and question novelty. Results from our ongoing research show that AI can significantly increase all three.

On the second path, AI can help transform the conditions and settings where people work so that questions that spark change — what we call “catalytic” questions — can emerge. This pushes leaders out of their comfort zones and into the position of being intellectually wrong, emotionally uncomfortable, and behaviorally quiet and more reflective, all of which, it turns out, promotes innovative thinking and action.

Let’s look at how each path can lead to breakthrough ideas.

## Increasing velocity, variety, and novelty.

Partnering with AI to ramp up the velocity, variety, and novelty of questions requires companies to train algorithms to answer the basic, easy (yes/no) questions independently and to reveal deeply buried patterns in the data. When this foundation is laid, humans can start exploring the power of more context-dependent and nuanced questions that AI technologies are not yet capable of answering alone.

## Question velocity

Algorithms can provide immediate answers to questions that leaders pose, in turn allowing them to ask more — and more frequent — questions. In our research, we found that 79% of respondents asked more questions, 18% asked the same amount, and 3% asked fewer.

At the cybersecurity firm Cybereason, researchers rely on AI and machine learning to immediately answer the basic questions about what happened in an apparent breach so the team can more quickly turn its attention to formulating deeper questions about why it happened. In the past, CEO Lior Div said, findings were more black-and-white: “It’s an attack. It’s not an attack. It’s good or it’s bad.” But the speed with which AI filled in those blanks opened up a whole new line of questions around intent — and what hackers are really after in a given situation.

Of course, there are risks to using AI to generate rapid-fire questions. For one, people may keep asking more and more questions without working their way toward an actionable path, making it important to recognize when the process stops being productive. For another, more questions don’t necessarily amount to better questions, which means you’ll still need to exercise human judgment in deciding how to proceed.

## Question variety

AI helps uncover patterns and correlations in large volumes of data — connections that humans can easily miss without the technology. Knowing they have this tool at their disposal frees up leaders to ask farther-ranging questions and explore new ideas that they may not have otherwise considered. In our research, we found that engagement with AI led respondents to ask different questions than they otherwise would have 94% of the time.

Consider this example: Kli Pappas, the director of predictive analytics at Colgate-Palmolive, told us that his team tapped AI to understand how charcoal became a wildly popular ingredient in consumer products so they could “find the next charcoal.” Their algorithm generated and answered thousands of questions based on their initial search for data, sketching out a decades-long trajectory from charcoal scrubs in South Korea 20 years ago to charcoal appearing in face washes in the U.S. and then in all kinds of products around the world. The AI-generated data led the team to ask hundreds of less-obvious questions to spark creative thinking about future trends that may be lurking in unexpected places. “We look backwards across categories and try to see how do trends move between categories from hair care, to skincare, to oral care,” Pappas said. “Just doing that puts you a decade or more ahead of the curve.”

## Question novelty

AI also facilitates deeper insights by helping users arrive at novel, “category jumping” questions — the gold standard of innovative inquiry — that apply understanding from one area to a completely different space. Our research shows that AI led respondents to ask unique questions that changed the direction of their team, organization, or industry 75% of the time.

When you know a technology can sift through much more data, and connect more dots, than you could ever do alone, it gives you license to ask wilder questions — things you would never ask if you had to answer them on your own, because they are intractable for the human brain or somehow go against entrenched cognitive biases.

While category-jumping questions will not arise in every encounter with AI systems, being open to the possibilities and allowing for freedom of inquiry can pave the way for more instances. Here’s how Mir Imran, a medical innovator and founder of InCube Labs, described the upside when we spoke: “AI can take really obscure variables and make novel connections. When these hidden connections come together, it causes you to reframe your question and deliver disruptive innovations.” In other words, AI’s novel connections can spark your novel questions, which in turn can lead you to investigate solutions others haven’t dreamed of yet — like the robotic pills that Imran’s team recently created to replace external injections with internal ones.

## Creating conditions for better questions.

AI can take leaders out of their usual mode of operation and force them to cede control over where their questions will take them. That’s a good thing. Increased question velocity, variety, and especially novelty give facilitate recognizing where you’re intellectually wrong, and becoming emotionally uncomfortable and behaviorally quiet — the very conditions that, we’ve found , tend to produce game-changing lines of inquiry. Jeff Wilke — former CEO of Amazon Consumer Worldwide, now a cofounder of Re:Build Manufacturing — has embraced these conditions not only in his day-to-day work as a tech executive but also throughout his career, continually revising his mental models while moving from role to role. When we spoke, he had this to say: “If you seek out things that you don’t know, and you have the courage to be wrong, to be ignorant, to have to ask more questions and maybe be embarrassed socially, then I think you build a more complete model, and that model serves you well over the course of your life.”

But there’s a hitch to teaming up with AI: Research suggests that it can be challenging for people to do so congenially because AI’s superhuman capabilities and unpredictable moves may prevent them from fully trusting and engaging with the technology. That tracks with what we’ve observed in organizations and learned from our conversations with leaders.

Distrust of the technology is hardly conducive to creative inquiry. So, look for ways to offset that, and don’t just leave it to AI to produce the conditions for breakthrough thinking and problem-solving. Consider how else you might create them. Where is there room in your problem-solving processes for synthesizing things that don’t seem related? How might you use those opportunities to throw people off balance so they’ll generate questions that reach beyond what they intellectually know to be right, what makes them emotionally comfortable, and what they are accustomed to saying and doing? At the same time, how can you create psychological safety for people in your organization to ask far-ranging questions and to use AI more effectively to learn from them, ultimately leading to asking better questions? When psychological safety is present, people can say, without repercussion, “I am wrong,” “I am uncomfortable,” and “I am still thinking”?

Rather than neatly resolve all those tensions, leaders and teams must learn to sit with the uncertainty that comes from asking questions that take them into new territory. While the process isn’t easy, the results are exciting, which is perhaps the most important benefit of collaborating with an AI system. Excitement provides momentum and motivation to push through a tough process, fueling further creativity.

## Mitigating AI’s Weaknesses with Human Strengths

Artificial intelligence may be superhuman in some ways, but it also has considerable weaknesses. For starters, the technology is fundamentally backward-looking, trained on yesterday’s data — and the future might not look anything like the past. What’s more, inaccurate or otherwise flawed training data (for instance, data skewed by inherent biases) produces poor outcomes.

Leaders and their teams must manage such limitations if they are going to treat AI as a creative-thinking partner. How? By focusing on areas where the human brain and machines complement one another. Whereas AI increases the volume of data we can process and the degree of complexity we can manage, our brains work in a reductive manner; we generate ideas and then explain them to other people. Whereas machines lack imagination and moral judgment, we can tap those critical skills as AI helps us increase the velocity, variety, and novelty of the questions we’re asking to solve problems in our organizations. Such differences are the stuff of fruitful collaboration — and optimizing them can reduce the threat of AI to human labor.

With humans and AI working to their respective strengths, they can transform unknown unknowns into known unknowns, opening the door to breakthrough thinking: logical and conceptual leaps that neither could make without the other. Harnessing this potential will require leaders to look at artificial intelligence in a new light — one that is less about cost savings, efficiency, and automation and more about inspiration, imagination, and innovation. It will also require building a culture that supports, incentivizes, and rewards asking big questions — and not necessarily knowing the answers.

- Hal Gregersen is a Senior Lecturer in Leadership and Innovation at the MIT Sloan School of Management , a globally recognized expert in navigating rapid change, and a Thinkers50 ranked management thinker. He is the author of Questions Are the Answer: A Breakthrough Approach to Your Most Vexing Problems at Work and in Life and the coauthor of The Innovator’s DNA: Mastering the Five Skills of Disruptive Innovators .
- NB Nicola Morini Bianzino is EY Global Chief Technology Officer, focused on bringing technology products to EY clients, positioning technology at the heart of the organization, advising global clients on technology investment and their innovation agendas, and providing industrialized technology products to meet their most pressing business needs. An early AI pioneer, he wrote a thesis on the application of neural networks to business in 1997. He holds a master’s degree in Artificial Intelligence and Economics from the University of Florence.

## Partner Center

Teach yourself statistics

## Statistics Problems

One of the best ways to learn statistics is to solve practice problems. These problems test your understanding of statistics terminology and your ability to solve common statistics problems. Each problem includes a step-by-step explanation of the solution.

- Use the dropdown boxes to describe the type of problem you want to work on.
- click the Submit button to see problems and solutions.

Main topic:

Problem description:

In one state, 52% of the voters are Republicans, and 48% are Democrats. In a second state, 47% of the voters are Republicans, and 53% are Democrats. Suppose a simple random sample of 100 voters are surveyed from each state.

What is the probability that the survey will show a greater percentage of Republican voters in the second state than in the first state?

The correct answer is C. For this analysis, let P 1 = the proportion of Republican voters in the first state, P 2 = the proportion of Republican voters in the second state, p 1 = the proportion of Republican voters in the sample from the first state, and p 2 = the proportion of Republican voters in the sample from the second state. The number of voters sampled from the first state (n 1 ) = 100, and the number of voters sampled from the second state (n 2 ) = 100.

The solution involves four steps.

- Make sure the sample size is big enough to model differences with a normal population. Because n 1 P 1 = 100 * 0.52 = 52, n 1 (1 - P 1 ) = 100 * 0.48 = 48, n 2 P 2 = 100 * 0.47 = 47, and n 2 (1 - P 2 ) = 100 * 0.53 = 53 are each greater than 10, the sample size is large enough.
- Find the mean of the difference in sample proportions: E(p 1 - p 2 ) = P 1 - P 2 = 0.52 - 0.47 = 0.05.

σ d = sqrt{ [ P1( 1 - P 1 ) / n 1 ] + [ P 2 (1 - P 2 ) / n 2 ] }

σ d = sqrt{ [ (0.52)(0.48) / 100 ] + [ (0.47)(0.53) / 100 ] }

σ d = sqrt (0.002496 + 0.002491) = sqrt(0.004987) = 0.0706

z p 1 - p 2 = (x - μ p 1 - p 2 ) / σ d = (0 - 0.05)/0.0706 = -0.7082

Using Stat Trek's Normal Distribution Calculator , we find that the probability of a z-score being -0.7082 or less is 0.24.

Therefore, the probability that the survey will show a greater percentage of Republican voters in the second state than in the first state is 0.24.

See also: Difference Between Proportions

## Every Number Counts: The Importance of Applied Statistics in Our Daily Lives

In today’s data-driven world, the application of statistics in everyday life is an ever-present reality that touches all aspects of society. Though the field of statistics originated centuries ago, the impact has exploded in recent years as modern statisticians have advanced applications of statistics through innovative, problem-solving approaches.

This blog will explore contemporary uses of statistics in everyday life, and the infographic following highlights vital examples.

See the infographic version

## The Evolution of Statistics in the Real World

The technologies powering many of the products we buy, shows we watch, and devices we use today were developed and perfected through the efforts of mathematicians, demographers and statisticians — long before tech companies entered the picture.

The idea of a census is a prime example of statistical progress over time. Historically, governments have used censuses to track population size. For instance, English demographers John Graunt and William Petty applied mathematical techniques to estimate population changes in the 1600s. In the United States, the first census dates back to 1790.

The process and ramifications of the U.S. census have continued to expand, covering an increasingly broad range of demographic and economic information. Today’s U.S. census determines vital areas of government that significantly impact daily life, such as allocation of public funding, congressional representation and delineation of school districts.

The role of statistics in the real world extends far beyond the census, however. The federal government now operates 13 statistical agencies that manage critical information related to labor trends, health, education and more. Statistics also influence the operations of industries, markets and even nonprofits.

## The Scope of Statistics in Everyday Life

The practice of applied statistics plays a role in every realm of life today. The application of statistics most often happens in the background, as statisticians are continuously at work to discover and implement world-shaping developments.

Applying statistics in the real world extends to every aspect of government in countries around the world. The United Nations Statistics Division describes the role of official statistics as an “indispensable element in the information system of a democratic society.”

Politicians and campaign managers use statistics to target specific voter demographics, gauge rates of constituent approval, and predict elections. Additionally, law enforcement agencies track data about fraud and crime that is then used to evaluate the effectiveness of strategies and tactics.

Government statistics also affect daily life in many less obvious ways. These are some examples from the U.S. government of statistics in everyday life:

- Economic numbers related to production, investment and trade affect financial policies and taxes. Governments, market leaders and other statisticians depend on this information to understand how the national economy is performing and how this affects their interests.
- Federal science research advances scientific and engineering discovery and integrates this work into education. Leading-edge federal research supports the nation’s security and international leadership.
- The collection and analysis of educational data inform leaders on key indicators concerning the condition of education. The scope of information includes findings in areas such as technological trends, public health, and educational methods.

## Health Care

Statisticians are big participants in pharmacology, as they’re involved with the discovery, testing, approval and marketing of a drug. They may also work in public health for government agencies, where they help to educate on community health matters and to develop preventative treatments and control.

Additionally, statisticians often take on roles in epidemiology, working in fields like nutrition and environmental science to help monitor and report on health-related data. For the World Health Organization, statistical data is considered a “core WHO activity” essential for advocacy and delivery of health initiatives.

Statistics often inform the development of legislation and may also guide in the interpretation of laws. Statisticians may provide expert testimony to court cases involving details such as salary discrepancies, DNA testing, disease clusters, and consumer surveys.

Statistical organizations serving state and federal courts synthesize information that serves to guide policy and procedural matters. For example, the National Center for State Courts has recently addressed questions related to the collection of race and ethnicity data . Data projects like this equip legal systems with critical information to promote equity and accountability.

Applications of statistics affect finance at many levels on a personal and global scale. Individuals use statistics to make decisions in financial planning and budgeting, while organizations are guided by statistics in financial policy decisions.

Banks use statistics to lower risk in lending operations, analyze activity in the financial market, and predict the impact of economic crises. Investors also use stats to understand the risk and potential of certain stocks, which helps them make informed investing decisions.

## Digital Marketing

Applied statistics is a driving force in transforming contemporary marketing approaches. The advent of “big data” means that companies are collecting phenomenal amounts of information from consumers. Proactive companies utilize this information to predict sales, glean customer interests, and analyze the effectiveness of marketing initiatives. Applying statistics in everyday life provides a highly targeted, data-driven strategy.

Advertisers in the form of paid search managers monitor ad campaigns based on key performance indicator targets and baselines and analyze data to continually optimize a campaign’s performance. SEO specialists guide companies in understanding how to read and interpret website analytics.

## Social Media Analytics

The rise of social media has created an environment where huge numbers of people and organizations are connected in a complex technological framework. In the Netflix film “The Social Dilemma,” statistics experts share what goes on behind social media screens. Everything users are doing online is “being watched,” the documentary explains.

Social media managers monitor organic and paid traffic to social media profiles and analyze data to grow followers, increase engagement and drive conversions. Companies use the data for microtargeting, measuring trends and watching competitors.

Learn More About Our Applied Statistics Program

## What Are Applications of Statistics for Your Career?

The role of statistician is one of today’s fastest-growing professions and rated by U.S. News & World Report as #6 in its “100 Best Jobs” list.

According to the Bureau of Labor Statistics, the number of positions for statisticians in the United States is expected to increase by 35% from 2019 to 2029. Salary is commensurate with demand, and the latest BLS numbers show that statisticians earned a median salary of $91,160 in 2019.

Job responsibilities for a statistician include:

- Using statistics to solve problems
- Analyzing and interpreting data
- Developing mathematical and statistical theories

Statistics careers can be found in almost any industry , covering a wide range of positions. Some of today’s top jobs include:

- Statistician
- Data scientist
- Computer and information research scientist
- Senior data analyst
- Data engineer
- Business intelligence analyst
- Senior financial analyst
- Statistics professor

## What Are Educational Requirements for a Statistician?

Though stats are used across many fields, specialized positions are only open to individuals with an education in mathematics or statistics, with most requiring a master’s degree. Professionals who earn a master’s degree in applied statistics gain a deeper understanding of how statistical solutions are applied in an organizational context for any industry.

The Michigan Tech online Master of Science in Applied Statistics equips students with expertise such as:

- Advanced statistical methods like predictive modeling, statistical data mining, parametric estimation, model diagnostics and forecasting.
- Integration of statistical tools into emerging technologies.
- Effectively communicating results of statistical analysis.

The MTU applied statistics program is ideal for working professionals, with 100% online coursework. There is no application fee and no GRE/GMAT requirement. There are three start dates each year, and the program is completed with 10 seven-week courses.

## How Will You Apply Statistics in Everyday Life?

The importance of data analysis and applied statistics is relevant to nearly every area of our lives. As the field of applied statistics continues to evolve, professionals qualified to lead organizations and governments with data insights will make a significant impact on the lives of generations to come.

Take the next step toward your Master of Science in Applied Statistics at Michigan Tech.

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In today’s data-driven world, the application of statistics in everyday life is an ever-present reality that touches all aspects of society. Learn more in this infographic from <a href=" https://onlinedegrees.mtu.edu ">Michigan Tech Online</a>.<br /><br /><a href="//onlinedegrees.mtu.edu/news/every-number-counts-importance-applied-statistics-our-daily-lives-infographic"><img style="width:100%;" src="//onlinedegrees.mtu.edu/sites/default/files/MTU-statistics-in-everyday-life-infographic.png"></a>

This article is adapted from one originally published March 15, 2019.

## The Role of Statistics in Computer Science

The role of statistics in computer science has evolved over the past decade and continues to play a critical part in developing and implementing data-driven technologies.

## What’s the Difference Between Data Science and Applied Statistics?

The value of data and professionals with data expertise is growing exponentially.

## Why is Statistics Important in Decision-Making?

Effective decision-making is crucial to the success of any business or organization.

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## Effects of problem-based learning modules within blended learning courses in medical statistics – A randomized controlled pilot study

Affiliation University of Belgrade, Faculty of Medicine, Belgrade, Serbia

Roles Data curation, Software, Validation, Writing – review & editing

Affiliation Faculty of Medicine, University of Pristina in Kosovska Mitrovica, Kosovska Mitrovica, Serbia

Roles Data curation, Methodology, Software, Writing – review & editing

Roles Methodology, Supervision, Validation, Writing – review & editing

Roles Resources, Supervision, Validation, Writing – review & editing

Roles Resources, Supervision, Visualization, Writing – review & editing

Affiliation Faculty of Medicine, University of East Sarajevo, Foca, The Republic of Srpska, Bosnia and Herzegovina

Roles Project administration, Resources, Supervision, Validation, Writing – review & editing

- Zoran Bukumiric,
- Aleksandra Ilic,
- Mirjana Pajcin,
- Dragana Srebro,
- Sasa Milicevic,
- Dragan Spaic,
- Nenad Markovic,
- Aleksandar Corac

- Published: January 26, 2022
- https://doi.org/10.1371/journal.pone.0263015
- Peer Review
- Reader Comments

Problem-based learning (PBL) allows students to learn medical statistics through problem solving experience. The aim of this study was to assess the efficiency of PBL modules implemented in the blended learning courses in medical statistics through knowledge outcomes and student satisfaction. The pilot study was designed as a randomized controlled trial that included 53 medical students who had completed all course activities. The students were randomized in two groups: the group with access to PBL modules within the blended learning course (hPBL group) and the group without access to PBL modules–only blended learning course (BL group). There were no significant differences between the groups concerning socio-demographic characteristics, previous academic success and modality of access to course materials. Students from hPBL group had a significantly higher problem solving score (p = 0.012; effect size 0.69) and the total medical statistics score (p = 0,046; effect size 0.57). Multivariate regression analysis with problem solving as an outcome variable showed that problem solving was associated with being in hPBL group (p = 0.010) and having higher grade point average (p = 0.037). Multivariate regression analysis with the medical statistics score as an outcome variable showed the association between a higher score on medical statistics with access to PBL modules (p = 0.045) and a higher grade point average (p = 0.021). All students in hPBL group (100.0%) considered PBL modules useful for learning medical statistics. PBL modules can be easily implemented in the existing courses within medical statistics using the Moodle platform, they have high applicability and can complement, but not replace other forms of teaching. These modules were shown to be efficient in learning, to be well accepted among students and to be a potential missing link between teaching and learning medical statistics. The authors of this study are planning to create PBL modules for advanced courses in medical statistics and to conduct this study on other universities with a more representative study sample, with the aim to overcome the limitations of the existing study and confirm its results.

Citation: Bukumiric Z, Ilic A, Pajcin M, Srebro D, Milicevic S, Spaic D, et al. (2022) Effects of problem-based learning modules within blended learning courses in medical statistics – A randomized controlled pilot study. PLoS ONE 17(1): e0263015. https://doi.org/10.1371/journal.pone.0263015

Editor: Gwo-Jen Hwang, National Taiwan University of Science and Technology, TAIWAN

Received: July 9, 2021; Accepted: January 7, 2022; Published: January 26, 2022

Copyright: © 2022 Bukumiric et al. This is an open access article distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Data Availability: All relevant data are within the manuscript and its Supporting Information files.

Funding: The author(s) received no specific funding for this work.

Competing interests: The authors have declared that no competing interests exist.

## Introduction

Medical doctors can be exceptional in their fields even if they do not know medical statistics, but they will be better if they do [ 1 ]. The study of Swift et al [ 2 ] showed that medical doctors considered medical statistics useful for “accessing clinical guidelines and evidence summaries, explaining risk levels to patients, assessing medical marketing and advertising material, interpreting the results of a screening test, reading research publications for general professional interest, and using research publications to explore non-standard treatment and management options”. Future physicians also thought that there was a need for a practical application of knowledge in medical statistics, not only its’ theoretical basis [ 3 ]. Lack of knowledge in medical statistics can lead to misinterpretation of clinical findings [ 4 ]. Statistical softwares, widely available now, enable an easy and comfortable analysis, but mistakes can be made when choosing the appropriate statistical test or assumptions for its’ application [ 5 ]. Medical students state that learning medical statistics through real life problems and the process of drawing conclusions can be more productive than traditional learning and knowledge assessment [ 6 ].

The process of education in medical sciences is most commonly based on traditional classroom lectures (face-to-face, lecture-based). In the past decade, there has been an increasing number of studies aiming to test, improve and introduce other forms of teaching, such as e-learning, blended learning, problem-based learning, team-based learning and flipped classroom [ 7 – 9 ].

A need to modify traditional classroom learning became a focal topic during the COVID-19 pandemic when teaching activities at universities worldwide were forced to shift to different types of online learning. It appears that in the future this challenge will be permanently changing the methods of physicians’ education [ 10 ]. The findings of new possibilities to transfer knowledge and skills through online learning modules and its’ constant improvement are receiving almost universal attention. In accordance with this, the implementation of problem-based learning in the online environment has shown similar success among students compared to in-person problem-based learning [ 11 ].

Problem-based learning enables students to learn through problem-solving experience [ 12 ]. During the learning process students’ main focus is on understanding and solving problems, rather than memorizing facts. Students develop critical thinking and clinical reasoning in concrete medical situation, which is very important for physicians’ daily practice [ 13 ]. There are positive experiences of combining problem-based learning (PBL) with lecture-based learning (LBL) and blended learning in education of future health care professionals [ 14 , 15 ]. A combined model of PBL + LBL was shown to be efficient in increasing the knowledge score, skills score and students’ satisfaction. This hybrid approach in learning has been increasingly used in Chinese medical faculties recently [ 16 ].

To the best of our knowledge, there are no previous studies on problem-based blended learning method for teaching medical statistics.

The aim of this study was to evaluate the effectiveness of implemented problem-based modules within blended learning courses in medical statistics through the outcomes of knowledge and student satisfaction. We created problem-based modules in medical statistics, based on actual problems which contained all of the steps in statistical analysis (defining the problem, choosing and applying adequate statistical tests, interpreting the results and drawing conclusions) and implemented them within the blended learning course.

## Materials and methods

The study was designed as a randomized controlled trial that included third-year medical students at the Faculty of Medicine, University of Pristina, Kosovska Mitrovica. The final analysis included 53 students who had completed all course activities out of 62 students who had been initially included in the study. Students were randomized in two groups: the group with access to problem-based modules within the blended learning curriculum (hybrid problem-based learning group–hPBL group) and the group with no access to problem-based learning modules–only blended learning course (blended learning group–BL group). The study began on October 1 st , 2019, at the beginning of the academic year, and was completed at the end of the academic year (September 30 th , 2020). As the research took part during the entire academic year, a part of the study was conducted during the COVID-19 pandemic. Classes in medical statistics and informatics were organized as a blended learning module.

Problem-based modules were conceptualized as an addition to the existing theoretical and practical curriculum in medical statistics within the blended learning course. PBL modules were created based on the technical solution verified by the board of the Ministry of Education, Science and Technological Development of the Republic of Serbia [ 17 ].

Blended learning course in medical statistics and informatics is based on a Moodle platform and contains 15 classes of theoretical lectures, 30 classes of practical exercises and 15 classes of other type, such as online readings or seminars. Total of 70% of the program of this course is comprised of medical statistics and this part of the course contains units on data types, descriptive statistics, confidence interval, probability and probability distributions, hypotheses testing, correlation and linear regression. Practical exercises are done using the statistical software Easy R (EZR) [ 18 ]. Students from both groups in our study had access to identical course activities (lectures and exercises), except for the access to the problem-based modules that were available only the students from hPBL group ( Table 1 ). During the course, students receive grades for all existing activities: lectures, exercises, colloquium, seminars, solving problems and final test. Students can see all the points for each activity any time during the course. The maximal number of points is 100 (70 for statistics and 30 for informatics). Students need to obtain the minimum of 51 points to pass the course. Based on the total number of points (51–100) the passing grades students can receive vary from 6 to 10.

- PPT PowerPoint slide
- PNG larger image
- TIFF original image

https://doi.org/10.1371/journal.pone.0263015.t001

Our study examined only the outcomes of medical statistics (70% of the course): problem solving score (5 problems with maximum score of the total of 25 points) and total medical statistics score (theoretical knowledge score, practical exercises score, problem solving score, independent students’ assignments score, seminars and colloquium; the maximal total medical statistics score was 70 points). The final grade in the course could not be compared because of the score in medical informatics, since the course contains both medical statistics and medical informatics. An anonymous online questionnaire using the five-point Likert scale (1 point- low satisfaction, 5 points- high satisfaction) was used to assess the students’ satisfaction in hPBL group with the PBL modules.

PBL modules were created based on the structure of the steps in statistical analysis ( Fig 1 ). Statistical analysis of the research problem is based on the multiple successive steps which include the following: the definition of the problem and the research question, recognition of the data type, sample type and hypothesis, selection of the adequate statistical test, application of the test, interpretation of the results and conclusion related to the description of the data, statistical conclusion and implications of the results. The PBL modules use guiding questions following the steps of statistical analysis. The guiding questions consisted of: interactive multiple choice or open-ended questions and followed a similar principle to the one Brown et al. applied [ 19 ]. Guiding questions changed within each step, based on the type of the statistical problem, number of variables and the sample (examples can be seen online following the link provided in the text below). Students can understand the necessary components for statistical reasoning by answering these questions and learn how to solve the problem.

https://doi.org/10.1371/journal.pone.0263015.g001

PBL modules were incorporated in each unit within the blended learning curriculum in Medical statistics and informatics for students in hPBL group. For each unit, students had PBL modules to resolve and to synthesize knowledge from theoretical and practical modules. Each PBL module was created and moderated by the teacher (tutor in classical problem-based learning). All the necessary information was given to students during the lectures, while the exercises and materials were given in the blended learning course. Students can resolve the problems alone or in communication with other students within the group (sending messages on Moodle platform, by asking questions in the forum discussion specially designed for these purposes, or by addressing questions to the online moderator- teacher). Students can use all available materials from the blended learning course, as well as other online materials or books during problem solving. The success in each step is evaluated. After completing all the necessary steps, students receive points and depending on the aim of the module, they also receive correct answers. The system supports the possibility of repeating an attempt of solving the problem until the student achieves the minimal necessary knowledge level, or desired knowledge level. PBL module contains meta-cognitive characteristics such as planning, managing and application of the previously adopted knowledge.

Examples of PBL modules used in this study are based on the course Problem based modules in medical statistics and can be accessed via the following link: http://e-ucenje.med.pr.ac.rs/course/view.php?id=232 (username: user; password: user).

## Ethical statement

The study was approved by the Ethical Committee of the Faculty of Medicine, University of Pristina, Kosovska Mitrovica (No. 09–3171). During the first week of the course (the first week of the semester), before the randomization in the groups was performed, students received written and oral explanations of the study, the processes and aims, the modalities of data gathering and data analysis. Students were explained that all the information gathered would be anonymous, that the participation was voluntary and that they could dropout of the study at any point. After this, the students gave an oral consent for their participation in the study, which was then verified in their records. A questionnaire on satisfaction was filled in after the course had been completed, as an online anonymous and non-obligatory questionnaire. All the data on the course outcomes and the data from the questionnaire on the students’ satisfaction were gathered by the administrator of the Moodle platform who was the only one with the access to the complete database. The authors of the study only had access to the anonymized database that is provided with the manuscript.

## Statistical analysis

Based on the variable types and normality of distribution, description of the data was shown as number (n) and percentage (%), mean±standard deviation or median (range, minimum- maximum). T-test, Mann-Whitney test, Chi-square test or Fisher’s exact test were used to test the hypotheses. The effect size in t-test was examined with Cohen’s d . Linear regression was used to analyze the learning outcome (problem solving score and total medical statistics score) and its potential predictors. All the variables which were significant in the univariate models at the level of 0.05 were entered in the multivariate regression analyses. Statistical hypotheses were tested at the significance level (alpha) of 0.05.

There were no statistically significant differences between the students from the hPBL and BL group concerning socio-demographic characteristics, grade point average and the modality of access to the materials within the blended learning course ( Table 2 ).

https://doi.org/10.1371/journal.pone.0263015.t002

Students in hPBL group had a significantly higher problem solving score (p = 0.012, effect size 0.69) and total medical statistics score (p = 0.046, effect size 0.57) ( Table 3 ). Students in hPBL group had a significantly higher score on total satisfaction with the course (median 5, range 4–5) compared to the students in BL group (median 5, range 3–5), (p = 0.012).

https://doi.org/10.1371/journal.pone.0263015.t003

Multivariate regression analysis with problem solving as an outcome variable showed that problem solving was associated with being in hPBL group (p = 0.010) and having higher grade point average (p = 0.037) ( Table 4 and Fig 2 ).

hPBL –hybrid Problem Based Learning, BL –Blended Learning.

https://doi.org/10.1371/journal.pone.0263015.g002

https://doi.org/10.1371/journal.pone.0263015.t004

Multivariate regression analysis with total medical statistics score as an outcome variable showed that the total medical statistics score was associated with hPBL group (p = 0.045) and higher grade point average (p = 0.021) ( Table 5 and Fig 3 ).

https://doi.org/10.1371/journal.pone.0263015.g003

https://doi.org/10.1371/journal.pone.0263015.t005

All the students in hPBL group (100.0%) thought that the PBL modules helped them to achieve the desirable knowledge in medical statistics. On the five-point Likert scale (1 –the lowest satisfaction, 5 –the highest satisfaction) median grade for adequacy of the modules, modalities of solving problems, the assistance received in the process of learning medical statistics, the students graded problem solving modules with the median grade 5 (range 4–5). Median on the interest in problem solving modules was 4.5 (range 3–5) ( Table 6 ).

https://doi.org/10.1371/journal.pone.0263015.t006

In this study, it has been shown for the first time that the implementation of problem-based learning into blended learning course in undergraduate medical studies contributes to better learning outcomes in medical statistics. The results of this research indicate that students of the hPBL group had a significantly higher problem solving scores and total medical statistics scores. The presented PBL modules enable active learning of medical statistics by solving actual statistical problems, through conceptual understanding, and they direct students to logical thinking. This direction is in line with recommendations from the Guidelines for Assessment and Instruction in Statistics Education (GAISE) Reports published by the American Statistical Association (ASA) [ 20 ]. Also, our results are consistent with the results of a systematic review that relates to the benefits of problem-based learning over traditional learning in biomedical education [ 21 ]. Like in our study, medical students had better solving scores and satisfaction [ 21 ]. The results of another meta-analysis showed that combined PBL and LBL learning of clinical medicine was significantly superior in achieving higher knowledge and skills scores, as well as learning satisfaction compared to LBL alone [ 22 ]. This meta-analysis included studies from China, and its’ authors suggested hybrid PBL to be gradually introduced into clinical medical teaching programs [ 22 ] since the results from previous meta analysis in China had shown that the competencies of students in medical statistics were insufficient and that they did not have sufficient ability to practically apply their statistical knowledge [ 23 ].

Teacher in medical statistics acted as a tutor in our study, he/she was obligated to answer students’ questions and to evaluate the outcomes of the problem based learning. In the study of Woltering et al [ 24 ] blended problem-based learning included the e-learning module complementary to the classic PBL modules, but without the inclusion of tutors. This study showed an increase in students’ motivation, subjective gains in knowledge and overall satisfaction among students in blended program-based learning. There was no significant difference in the successfulness of problem solving, which confirms that the PBL can be successfully implemented in online learning environments. Additionally, in the study which compared traditional classroom classes and online asynchronous PBL, de Jong et al [ 25 ] found that the absence of a formal tutor can force students to rely on themselves and teamwork, develop critical thinking, analytical and self-regulation skills.

The results of our study suggest that the modality used to access the course is a significant predictor of the problem solving score (if it is via PC/laptop or Tablet/Smartphone). Students who accessed the course via PC/laptop had better scores, which was expected because the problems are solved in the EZR software via PC, and the access via smartphone or tablet does not allow access to this feature. On the other hand, access to theoretical materials may be more comfortable/practical via tablets. However, the method of access is not a statistically significant predictor of total medical statistics score.

Self-rated knowledge through the expected final grade was significantly associated with the total medical statistics score in univariate regression model, which was, most likely, influenced by multiple accesses to PBL modules and students’ motivation. The motivation was not directly assessed in our study, but was assessed indirectly, through the expected final grade, as there is a proven positive association between the intrinsic motivation and perceived academic rank [ 26 ]. It is expected that highly motivated students would use the advantages of an online platform more and would have higher scores in solving actual problems and higher total medical statistics score.

Multivariate regression models showed a significant association between the problem solving score and total medical statistics score with hPBL group and higher grade point average. We expected that students with a higher grade point average have higher problem solving as well as the total medical statistics score, regardless of the study group. Although we did not have a large sample and we did not find a great difference between the means of the two groups, the effect size was moderate. However, the implemented PBL modules can be adapted to teaching in medical statistics, as they offer a modern method of solving practical problems that is appealing to students. Such an approach of directing them through a problem, from its’ recognition to the analysis of results, helps students to better understand medical statistics and can help them later during their scientific research work. Depending on the curriculum and concept of teaching medical statistics, PBL can be used as an addition to theoretical and/or practical classes. PBL can be easily implemented in different teaching models (classic, online, blended), which in our case proved to be extremely useful during the COVID-19 pandemic. PBL modules can be used not only during classes for learning or updating materials, but also for practicing assignments and self-assessment of knowledge by students. The PBL is completely independent from the modality of calculation (classical or software) or on platform (can be solved in a classroom, on paper, or using an online study platform). The optimal application of PBL is within online courses because they allow students to access materials when it suits them, as often as they need to (self-regulated learning). Educational reform that combined Moodle with the traditional way of learning medical statistics has achieved good results among students [ 27 ]. The meta-analysis showed that the blended learning methods were more efficient than traditional learning in medical sciences [ 28 ]. Also, PBL can be used for assessment of knowledge on the exam itself. This is supported by the results from our study, which showed that the time of taking the exam (before or during the COVID-19 pandemic) was not a statistically significant predictor of problem solving score and total medical statistics score. During the COVID-19 pandemic, there were also changes in testing methods [ 29 ] and the possibility of introducing the open-book examination [ 30 ]. The applied PBL modules can also be used for online assessment of problem solving from medical statistics with the open book model, since in a limited time linking of all information needed for problem solving cannot be compensated by searching on the Internet, books or any study materials.

The potential for the application of these modules is high, in the context of the current level of presence of e-learning methods in medical sciences and especially, due to the increase in digitalization during the COVID-19 outbreak. The PBL can be used for learning, repetition, exercise, and self-assessment, and finally, for the assessment in the exam. This increases its usefulness threefold.

The results from the meta- analysis showed that the courses which implemented the PBL were associated with long term knowledge retention, short term retention and application of clinical skills and thinking. Traditional approach is more convenient for short term knowledge retention which does not require further understanding [ 31 ]. We did not examine the knowledge retention, and it could be included in the further research.

## Limitations

This study has a few possible limitations. Firstly, the study was conducted in only one educational institution, during one study course, with a small number of participants and this should be taken into account when generalizing the results in other courses, study programs or universities. Secondly, the PBL modules include only modules in the basic course of medical statistics. Although this concept and its technical solution can be applied to higher levels of education, there is a need for the assessment of its efficiency, which is our aim for further studies. Another potential limitation of the study we could not control was the fact that the students from the hPBL group could show the modules to the students in BL group. This risk can be minimized through the inclusion of a larger number of participants in the study and through the development of a larger number of the problems for students to solve. The enthusiasm of students and their reaction to the new type of learning, especially during the COVID-19 pandemic, can also affect higher grades among the hPBL group.

## Future research

The authors of this study are planning to create PBL modules for advanced courses in medical statistics and to conduct the study on other universities with a more representative study sample, with the aim to overcome the limitations of the existing study and confirm its results.

The presented PBL modules can be easily implemented in the existing courses of the medical statistics developed on the Moodle platform, have high applicability and can complement, but not replace other forms of teaching. The PBL modules allow students to associate theory and practice, synthesize the existing knowledge and generate new knowledge and show how medical statistics can be thought through conceptual understanding via directing students through problems using the step-by-step approach. The problem-based modules were shown to be efficient in acquiring knowledge, are well accepted among students and can be a missing link in learning and understanding medical statistics.

## Supporting information

S1 database..

https://doi.org/10.1371/journal.pone.0263015.s001

## Acknowledgments

This article is dedicated to our teacher and friend Professor Goran Trajkovic, who passed away prematurely.

- 1. Trajkovic G. “Introduction to medical statistics”. Lectures. Faculty of Medicine, University of Pristina in Kosovska Mitrovica. 2015.
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- 5. Masic I, Jankovic SM, Begic E. PhD Students and the Most Frequent Mistakes during Data Interpretation by Statistical Analysis Software. Studies in Health Technology and Informatics. IOS Press; 2019. pp. 105–109. https://doi.org/10.3233/SHTI190028 pmid:31349277
- 17. Bukumirić Z, Trajković G. Problem-based model for acquiring and assessment knowledge in statistics and informatics in medicine, statistical data analysis and interpretation of results. Technical solution—M85, ID 5114026, Project TR37016. Ministry of Education, Science and Technological Development of the Republic of Serbia; 2019.

## Learning over Knowing: Why You Need to Change Your Problem-Solving Practices with Dhiraj Rajaram Leaders of Analytics

This episode of Leaders of Analytics features Dhiraj Rajaram, the Founder of global decision sciences company Mu Sigma. Mu Sigma serves more than 140 of the Fortune 500 and the company’s mission is to simplify complex problems through the science of decisions. Dhiraj shares his views on problem-solving in business, and how Mu Sigma's three core beliefs have been instrumental in the company's success. At Mu Sigma, they believe in "Learning over Knowing", "Extreme Experimentation", and "The New IP". Their data-driven decision-making approach has helped solve some of the toughest business challenges and has set them apart from the competition. As an entrepreneur or business leader, you'll gain valuable insights into using data to solve complex issues, as well as an insider's perspective on Dhiraj's entrepreneurial journey. In this episode we discuss: Dhiraj’s entrepreneurial journey from a one-man band to leading thousands of employees The critical moments that led Dhiraj to become a founder and the key elements of entrepreneurial success Mu Sigma’s unique recruitment and training strategy What you can learn from Mu Sigma’s three core beliefs How to make better decisions for your organisation, and much more. Mu Sigma's website Connect with Dhiraj on Linkedin.

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- Copyright 2021 All rights reserved.

## How to Solve Statistics Problems Accurately

Several students are struggling with the problem of mathematics numeric problems. A study shows that almost 30% of students are unable to solve quantitative problems.

Therefore, in this blog, you will find effective solutions for how to solve statistics problems. Here you will find various advanced quantitative data analysis courses.

Because of the various uses of these statistics problems in everyone’s daily lives, students still lack solving these kinds of problems. That is why it becomes necessary to understand the methods to tackle the problem of statistics.

So, let’s check all the necessary techniques to solve quantitative data problems.

## What is statistics?

Table of Contents

It is one of the branches of mathematics statistics that involves collecting, examining, presenting, and representing data.

Once the information is accumulated, reviewed, and described as charts, one may see for drifts and attempt to execute forecasts depending on certain factors.

Now, you have understood the meaning of statistics. So, it is the right time to get familiar with the steps used for how to solve statistics problems.

Here, you will find out these techniques with a suitable example. This will help you to know how these techniques are implemented to solve quantitative statistics problems.

But before moving to the strategies, let’s check whether you have effective knowledge of statistics or not. This will also help you to check whether your concepts about the statistics problem are cleared or not.

Once you know that you have an effective understanding of statistics, you can easily solve the statistics problems.

Take a test of your statistics knowledge !!!

Give the answers to questions mentioned below:

- How long do seniors spend clipping their nails?
- Not statistical
- Statistical
- None of both
- How many days are in Feb?
- Did Rose watch TV last night?
- How many cyberspace searches do citizens have at a Retirement each day?
- How long is the rapunzel’s hair?
- The average height of a giraffe?
- How many nails does Alan have in his hand?
- How old is my favourite teacher?
- What does my favorite basketball team weigh?
- Does Morris have a university degree?

Now, you have tested your knowledge so we can move to the strategies to solve a statistical problem.

## Strategies for how to solve statistics problems

Let’s take a statistical problem and understand the strategies to solve it. The below strategies are based on the random sample problem and solve it sequentially.

## #1: Relax and check out the given statistics problem

When students assign the statistics problems, you have noticed that they get panicked. Due to panic, there are higher chances of making errors while solving statistics distributions.

This might be because students think that they can solve these queries, leading to low confidence. That is why it becomes necessary to calm yourself before you start to solve any statistics problem.

Here is an example that helps you to understand the statistics problem easily.

Almost 17 boys were diagnosed with a specific disease that leads to weight change.

Here the data after family therapy was as follows:

11,11, 6, 9, 14, -3, 0, 7, 22, -5 , -4, 13, 13, 9, 4 , 6, 11

## #2: Analyze the statistics problem

Once you assign the statistics problem, now analyze the query to solve it accurately.

Check what does it ask you to perform in the problem? It would help if one obtained the upper confidence limit that can utilize the mean: the degrees of freedom and the t-value.

Here is the question: what is the meaning of the degrees of freedom to a t-test?

Take a sample question: If there are n number of observations. It would help if you estimated the mean value. This will leave the n-1 degree of freedom that is utilized for estimated variability.

For the above problem, we can estimate the average along with the sample value 17-1 that is equal to 16.

To recognize the difficulty, study the numbers one can DO have.

- One should have a lower confidence limit.
- Get all of the specific scores.
- You need to understand the number of scores (17).

Consider the things about what one can DO remember (or may view within a textbook).

- The mean score of the number is the addition of the scores divided with the total score number.
- To get the lower confidence limit, one needs to do minus (t * standard error).
- An UPPER confidence limit is the collected average + (t * standard error).

## #3: Choose the strategy for how to solve statistics problems

There are several methods to get the upper confidence limit; besides this, all this includes the calculating value (t*standard error) to get the mean. There are the easiest approach is

- Determine what the mean does.
- Check the difference in the mean and the limit of lower confidence.
- Sum the number to the mean.

These are steps where most people get puzzled. This might be because of the three main reasons.

- The first one is that students are stressed out because of indulging in various academic studies.
- Secondly, learners do not have enough time to check the statistics problems and recognize what to do first.
- Thirdly, they do not rest a single minute and study the right approach.

We think that several students do not pay sufficient time on the initial three levels before skipping to the fourth number.

## #4: Perform it right now

Take out a strategy.

- The mean will be 7.29.
- 7.29 -3.6 = 3.69
- Sum 3.69 to 7.29 to get 10.98

This is the correct answer.

## #5: Verify the to know how to solve statistics problems

Do a certainty verification. The mean must be 7.29. If it does not lay in the category of lower and upper confidence limits, then there would be something wrong.

Check again tomorrow to get the verification of the number. These steps would be implemented to all statistics problems (and a math query – might be a puzzle in life.)

Let’s understand the above steps by solving a statistical problem!!

Problem: In a state, there are 52% of voters Democrats, and almost 48% are republicans. In another state, 47% of voters are Democrats, and 53% are Republicans. If the sample takes 100 voters, then what probability represents the maximum percentage of Democrats in another state.

Solution:

P1 = Republican voters proportion in the first state,

P2 = Republican voters proportion in another state,

p1 = Sample Republican voters proportion in the first state,

p2 = Sample Republican voters proportion in another state,

n1 = Number of voters in the first state,

n2 = Number of voters in another state,

Now, let’s solve it in four steps:

- Remember that the sample size must be bigger to model difference for a normal population. Therefore, P1*n1 = 0.52*100 =52, (1-P1)*n1 = 0.48 *100 = 48.

On the other hand, P2*n2 = 0.47*100 =47, (1-P2)*n2 = 0.53*100 = 53, which is greater than 10. So we can say that sample size is much larger.

- Calculate the mean of the sample proportions difference: E(p1 – p2) => P1 – P2 = 0.52 – 0.47 => 0.05.
- Calculate the difference of standard deviation.

σd = sqrt{[ (1 – P2)*P2 / n2 ] + [ (1 – P1)*P1 / n1 ] }

σd = sqrt{[(0.53)*(0.47) / 100 ] + [ (0.48)*(0.52) / 100 ] }

σd = sqrt ( 0.002491 + 0.002496 ) = sqrt(0.004987) = 0.0706

- Calculate the probability. The given problem needs to calculate the probability, which is p1 < p2.

This is similar to determining the probability, which is (p1 – p2) < 0. To calculate the probability, you must transform the variable (p1 – p2) in the z-score. The transformation will be:

z (base (p1 – p2)) = (x – μ (base (p1 – p2) ) / σd = (0 – 0.05)/0.0706 => -0.7082

- With the help of the Normal Distribution calculator of Stat Trek’s, you can calculate that the Z-scores probability that is being -0.7082 is 0.24.

That is why the probability shows a greater % of Republican voters within another/second state as compared to the first state, and it is 0.24.

## Conclusion

To sum up this post, we can say that we have defined the possible strategies about how to solve statistics problems. Moreover, we have mentioned the procedure for solving the statistics queries that help students solve mathematics in their daily lives.

Besides this, we have provided solutions with detailed examples. So that students can easily understand the techniques and implement them to solve statistics terms.

Analyzing these examples can allow the students to know the sequence of solving a statistics question. Follow the steps mentioned above to get the desired result of the problems and verify them accordingly. Learn and practice the initial rule to solve each problem of quantitative analysis effectively. Get the best statistics homework help .

## Frequently Asked Questions

What are the four steps to organize a statistical problem.

The Four-Step to organize the statistical problem:

STATE: The real-world or a practical problem. FORMULATE: Which is the best formula to solve the problem? SOLVE: Make relevant charts and graphs and practice the required calculations. CONCLUDE: Take the summary to set the real-world problems.

## What is a good statistical question?

A statistical problem can be solved by gathering useful data and checking where the variability is in the given data. For instance, there is variability in the collected data to solve the problem, “What does the animal weigh at Fancy Farm?” but not to solve, “What is the colour of Ana’s hat?”

## What is the most important thing in statistics?

The three basic components of statistics are determination, measurement, and modification. Randomness is considered one way to supply development, and it is another way to model variations.

## Related Posts

## How to Find the Best Online Statistics Homework Help

## Must Have Business Analyst Skills To Become Successful

## Step by Step Process of How to Solve Statistics Problems

“How to solve statistics problems?” is an obvious question students mostly search over the internet.

For many students, it is like a nightmare to solve statistics problems due to various reasons. In order to solve statistics problems correctly, practice is a primary requirement, and you should know how and where to collect data and analyze and interpret it to draw valuable information.

Putting the right formulas to solve the problems is equally important as collecting the data from authentic and reliable sources. If you collect data from random sources, you can not conclude from that data.

So, if you are one of those who are facing problems when solving statistics problems, we are here to assist you.

In this blog, we will provide you with a step by step process of how to solve statistics problems. We will also cover statistical terms and definitions of statistics.

## What is statistics?

Table of Contents

Statistics is considered as the science that deals with methods and tools of collection, analysis, Interpretation and presentation of data. Statistics is majorly used for research and study purposes as through stats we can make significant decisions. It deals with both quantitative and qualitative data and structured and unstructured data.

So everyone is scared of statistics and they always search how to solve statistics problems. The general method of solving statistics problems is to write your question then collect data required for solving such a question and lastly you are required to analyse such data and to draw conclusion.

## Statistical and non-statistical problem

Let’s know the difference between a statistical problem and non statistical problem.

Question1. How many states are there in India?

Question 2. In which state girl ratio is maximum in India?

What do you understand from these both questions?

Have you noticed any difference?

Let me explain you-

The major difference between these questions is that Question number 1 is non statistical and question number 2 is statistical.

What makes these problems statistical and non statistical?

Four thing or factors make a problem statistical and non statistical that are given below-

- Way to ask the question
- Role of data and its nature
- Way to examine the data
- Types of interpretation you bring from research

Hence the question first is simple and factual and its answer does not need any type of research and collection of data whereas second question need to collect data from all the states, analyze data, research is required and at last we can conclude that which state have the maximum ratio of girls.

## Terminologies used in statistics problems –

There are n number of terminologies which are used in statistics this is why it is said that statistics has its own language which you should command first. So if you are searching for How to solve statistics problems then firstly you have to learn the meaning of basic terms used in Statistics. Following are the most essential terms –

When we solve any statistics problem then we are required to collect data from the people who are affiliated with the given question. So we have to decide whom we want to study. Thus, in statistics, people or individuals you want to study or you are studying are called as population. In short, the group of people whom you are studying is the population.

If you understand the term population then it is very easy to learn samples. The sample is just a subset of the total population. For example your population has 10 individuals then each individual is a sample for your study.

The next term to learn on How to solve statistics problems as the name suggests it is the scope of the study. That is the quantitative characteristics of the population you are studying or testing. For example you want to know how many people use Colgate. Then this question is a parameter. So your population and sample and other required details will be based on such parameters.

## Descriptive Statistic

Next terminology to learn in How to solve statistics problems is Descriptive statistics.

When you analyze the data after determining the hypothesis and collection of data then you will get certain results on such study and such result is called descriptive statistics.

## Procedure – How to solve statistics problems

## Determine your Question

The first step to solve the statistics problem is to decide the problem that is the question or hypothetical test. Unless you know the question you can’t process with other steps because this step will decide the parameter and population for your study. This is why this is the first and foremost step in How to solve statistics problems.

## Collection of Data

Next step is to collect the data as per your hypothesis. Here you will decide the population and you can use different methodologies of collection of data like questionnaires OR survey etc. It is also a very important step because you can’t get true and correct results unless you have correct data.

## Analysis of data

By now you have collected the required data and also you have your hypothesis so your next step in How to solve statistics problems is to analyse the data accordingly. There are various tools to analyse the data like Microsoft Excel, Python, R, etc. So you must be skilled in data analysis.

## Interpretation of data

Next step in How to solve statistics problems is to interpret the data you have collected. Point to note here is always remember your questions while interpreting because data speaks a lot so you have to scrutinize in such a way that you can get desired results. After this step you will get the results of your study so lastly you will have to just present the data. And for presentation also there are n number tools and methods which you can use. So presentation of data shall also be up to mark so that you can analyze the data easily and speedily. You can present the data through pie charts, graphs or tables etc.

## Statistical formulas

Statistical problems are solved through statistical formulas so the technique to learn such formulas is to break them down. For example if you are solving mean, median or mode or standard deviation you shall be well versed with these formulas then only you can get correct results.

Let’s take a statistics problem and solve it.

Suppose there are 10 students in a class and we are asked to find out the average weight of students of that class. For this we need to know about the weight of individual students so that we can calculate the average weight of those students.

Average weight- We can calculate average weight with the mean formula

Mean = sum of all terms/ total terms

Hence the average weight of students is 47.8 kg

Mode = the frequent term in the list is known as the median.

In the above question 45 is mode because it is repeated three times.

Median= The central term is known as median. But in this question we have ten terms and we have two middle terms.

Now the mean of these two terms is median. But before that we have to arrange the values in any order.

35 40 43 45 45 45 54 55 56 60

Here 45+45/2= 90/2 = 45 is median

Hence to solve statistics problems you should know these formulas or tactics.

In this competitive world, data analysis is the key stream to earn more profits and to beat the competition. Statistics is used for the same as it is the type of science which deals with data analysis and much more. Many people struggle with How to solve statistics problems so this article is inclined to that only. In case, if you need any help with statistics assignment , then you can get the best help from our statistics assignment helper .

In this blog we have also differentiate between statistical and non statistical questions so that you can better understand what statistics problems require.

What are the types of Statistics?

Statistics is mainly of two types-

Descriptive- It just describe what the data shows

Inferential- It helps in generalization of data and draws valuable conclusions.

What are the components of a statistics problem?

- Ask a question
- Gather Data
- Data analysis
- Interpret Results

Which formulas should we know about how to solve statistics problems?

In statistics, we use numerous formulas to solve the different problems. Statistics problems require simple as well as complex formulas to give answers. We have to use formulas of mean, mode, median, and other probability formulas in statistics.

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## Computer Science > Computation and Language

Title: tree of thoughts: deliberate problem solving with large language models.

Abstract: Language models are increasingly being deployed for general problem solving across a wide range of tasks, but are still confined to token-level, left-to-right decision-making processes during inference. This means they can fall short in tasks that require exploration, strategic lookahead, or where initial decisions play a pivotal role. To surmount these challenges, we introduce a new framework for language model inference, Tree of Thoughts (ToT), which generalizes over the popular Chain of Thought approach to prompting language models, and enables exploration over coherent units of text (thoughts) that serve as intermediate steps toward problem solving. ToT allows LMs to perform deliberate decision making by considering multiple different reasoning paths and self-evaluating choices to decide the next course of action, as well as looking ahead or backtracking when necessary to make global choices. Our experiments show that ToT significantly enhances language models' problem-solving abilities on three novel tasks requiring non-trivial planning or search: Game of 24, Creative Writing, and Mini Crosswords. For instance, in Game of 24, while GPT-4 with chain-of-thought prompting only solved 4% of tasks, our method achieved a success rate of 74%. Code repo with all prompts: this https URL .

## Submission history

- Download a PDF of the paper titled Tree of Thoughts: Deliberate Problem Solving with Large Language Models, by Shunyu Yao and 6 other authors PDF
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Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs .

Missouri State University

Check it out

## Creative problem solving opened up new pathway for businesses to invest in employees

- Date published May 24, 2023

## Accounting firm builds strong international employee base through partnership with English Language Institute.

When presented with a problem, sometimes the solution means you pivot and do things even better than before. Pivot – that’s a term we all used and heard a lot during the COVID-19 pandemic. But creative problem solving really can turn a problem into the establishment of a best practice.

That’s what happened with the English Language Institute at Missouri State University early in the pandemic. Embassies were closed. Students experienced difficulties getting visas. And the ELI’s primary purpose – to teach American culture and improve English language skills to MSU international students – wasn’t in as high of a demand.

Paula Moore, like many others in the division of community and global partnerships at MSU, was tasked with thinking about alternative markets and pathways. Moore began considering how the ELI could provide outreach services to employers wanting to improve cultural relations with employees with English as a second language – both international students and refugees.

“They say two heads are better than one, especially if the second head has seen other parts of the world and different ways of doing things,” Moore said. “That creates multiple options for solving problems.”

## A partnership begins

In late 2021, Abacus, an entrepreneurial accounting firm, reached out for help. Abacus was looking for a way to better support international employees.

“We weren’t sure what it needed to look like, but we knew we could live out our firm’s foundational principle of ‘People Deserve Better,’” said Andrea Battaglia, Chief Excellence Officer at Abacus. “As we began to discuss ideas, we reached out to a contact at MSU that connected us to Paula, and the rest is history.”

Together, Moore and the team at Abacus configured and signed a contract for services to assist with the training and transition for their international employees.

“They’re looking at the potential of doubling the size of their company in the next two to three years, and they see working with international students as a great opportunity to fill the ranks there,” Moore said.

## More than learning a language

But Moore points out that the training has very little to do with language acquisition – these employees are often taught English from a very young age. Instead, it’s the idioms – like, “I’m all in” and “cut to the chase” and “hit it out of the park” – that come up in the discussions more often than not.

“Sometimes they also experience cultural adjustment issues, like what is the expectation of a good employee in the U.S.?” Moore said. “We discuss some of the differences in expectations in terms of taking initiative versus waiting to be told what you’re supposed to do and contributing during staff meetings. We’re just helping to raise awareness of where those disconnects might be happening. Then we try to illuminate how they can understand each other better.”

The program for Abacus has three MSU participants: Moore, Pascal Hamon and Terry Barakat, who meet monthly with the four international employees at Abacus. Then, each of those employees also meets monthly 1-on-1 with the advisors.

“This program surreptitiously is for the international employees, but the other function is to prepare the entire organization to embrace international team members and create a more welcoming place for them,” Moore said. “It’s really laying the groundwork for creating a more diverse workforce in that company.”

Interested in building something for your organization? Reach out! Contact [email protected].

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## COMMENTS

Consider statistics as a problem-solving process and examine its four components: asking questions, collecting appropriate data, analyzing the data, and interpreting the results. This session investigates the nature of data and its potential sources of variation. Variables, bias, and random sampling are introduced. View Transcript In This Session

2. Learning and Teaching Through Problem Solving A simple paradigm for solving problems using statistics is summarised in the English National Curriculum using four activities: specify the problem and plan; collect data from a variety of suitable sources; process and represent the data; and interpret and discuss the results.

A simple paradigm for solving problems using statistics is summarised in the English National Curriculum using four activities: specify the problem and plan; collect data from a variety of suitable sources; process and represent the data; and interpret and discuss the results.

The word statistics may bring to mind polls and surveys, or facts and figures in a newspaper article. But statistics is more than just a bunch of numbers: Statistics is a problem-solving process that seeks answers to questions through data. By asking and answering statistical questions, we can learn more about the world around us. Statistics is ...

Unit 7: Probability. 0/1600 Mastery points. Basic theoretical probability Probability using sample spaces Basic set operations Experimental probability. Randomness, probability, and simulation Addition rule Multiplication rule for independent events Multiplication rule for dependent events Conditional probability and independence.

A video series focused on bringing Steve's problem solving approach to life for students, via an audio/visual modality. Our community has ranked the five most difficult examples in every chapter, and these whiteboard-esque videos walk our learners through the application of the 4-step problem solving methodology used throughout the project.

Teaching about problem solving. begins with suggested strategies to solve a problem. For example, "draw a picture," "make a table," etc. You may see posters in teachers' classrooms of the "Problem Solving Method" such as: 1) Read the problem, 2) Devise a plan, 3) Solve the problem, and 4) Check your work. There is little or no ...

Free math problem solver answers your statistics homework questions with step-by-step explanations. Mathway. Visit Mathway on the web. Start 7-day free trial on the app. ... While we cover a very wide range of problems, we are currently unable to assist with this specific problem. I spoke with my team and we will make note of this for future ...

AI Can Help You Ask Better Questions — and Solve Bigger Problems. by. Hal Gregersen. and. Nicola Morini Bianzino. May 26, 2023. Paul Taylor/Getty Images. Summary. Most companies still view AI ...

Problem 1. In one state, 52% of the voters are Republicans, and 48% are Democrats. In a second state, 47% of the voters are Republicans, and 53% are Democrats. Suppose a simple random sample of 100 voters are surveyed from each state. What is the probability that the survey will show a greater percentage of Republican voters in the second state ...

Using task-based interviews of students who completed a second course in statistics, the framework was used to determine the ways in which students utilize statistical computing tools, specifically R, while going through problem solving phases. Patterns that emerged are discussed. Supplementary materials for this article are available online.

Though the field of statistics originated centuries ago, the impact has exploded in recent years as modern statisticians have advanced applications of statistics through innovative, problem-solving approaches. This blog will explore contemporary uses of statistics in everyday life, and the infographic following highlights vital examples.

Here, you will find all the help you need to be successful in your statistics class. Check out our statistics calculators to get step-by-step solutions to almost any statistics problem. Choose from topics such as numerical summary, confidence interval, hypothesis testing, simple regression and more. If you still need more help, visit our other ...

A Problem Solving ApproachNow available with Macmillan's Achieve, Introductory Statistics helps students develop the fundamental lifelong skill of solving problems and interpreting solutions in real-world terms. One of our goals was to make this problem-solving approach accessible and easy to apply in many situations. We certainly want students to appreciate the beauty of statistics and ...

Great Expectations: Probability Through Problems. The resources found on this page offer a new approach to teaching probability. The articles outline the thinking behind the approach, and explain the research basis for choosing to teach probability in this way. Then there are resources for teaching the curriculum - rich, hands-on classroom ...

Statistical Thinking for Industrial Problem Solving is an applied statistics course for scientists and engineers offered by JMP, a division of SAS. By completing this course, students will understand the importance of statistical thinking, and will be able to use data and basic statistical methods to solve many real-world problems.

With a clear and detailed approach to the fundamentals of statistical theory, Examples and Problems in Mathematical Statistics uniquely bridges the gap between theory andapplication and presents numerous problem-solving examples that illustrate the relatednotations and proven results.

Problem-based learning (PBL) allows students to learn medical statistics through problem solving experience. The aim of this study was to assess the efficiency of PBL modules implemented in the blended learning courses in medical statistics through knowledge outcomes and student satisfaction.

Steps of How To Solve Statistics Problems 1. Ask a Question 2. Collect Data 3. Analyze Data 4. Interpret Results BONUS POINT Common Problems When Using Statistics Removing Meaning Out Of Little Difference Use of Small Sample Sizes Poor Survey Design To Sum Up Frequently Asked Questions (FAQs) What are some examples of statistics in everyday life?

Mu Sigma serves more than 140 of the Fortune 500 and the company's mission is to simplify complex problems through the science of decisions. Dhiraj shares his views on problem-solving in business, and how Mu Sigma's three core beliefs have been instrumental in the company's success.

What is statistics? Strategies for how to solve statistics problems #1: Relax and check out the given statistics problem #2: Analyze the statistics problem #3: Choose the strategy for how to solve statistics problems #4: Perform it right now #5: Verify the to know how to solve statistics problems Conclusion Frequently Asked Questions

Statistical problems are solved through statistical formulas so the technique to learn such formulas is to break them down. For example if you are solving mean, median or mode or standard deviation you shall be well versed with these formulas then only you can get correct results. Let's take a statistics problem and solve it.

Language models are increasingly being deployed for general problem solving across a wide range of tasks, but are still confined to token-level, left-to-right decision-making processes during inference. This means they can fall short in tasks that require exploration, strategic lookahead, or where initial decisions play a pivotal role. To surmount these challenges, we introduce a new framework ...

In late 2021, Abacus, an entrepreneurial accounting firm, reached out for help. Abacus was looking for a way to better support international employees. "We weren't sure what it needed to look like, but we knew we could live out our firm's foundational principle of 'People Deserve Better,'" said Andrea Battaglia, Chief Excellence ...