Course title

MAT315

Pre-requisite

Passing grade in MAT100 IGCESE Mathematics I or MAT200 IGCSE Mathematics II

Course description

Course Description
This course teaches students how to use four-steps of the statistical process in the context of sports: ask questions; collect data; analyze data; and make conclusions. Each chapter will begin with a sports-related statistical question (e.g.; Is there a home field advantage in the NFL?) and then students will learn how to collect appropriate data; how to analyze the data; and how to make reasonable conclusions. Although the context of the examples and exercises will be sports related; the primary focus of the class will be to teach students the basic principles of statistical reasoning. Major statistical topics include: analyzing distributions of univariate and bivariate data; both categorical and numerical; using graphs and summary statistics; correlation and least squares regression; using simulations to estimate probability distributions; theoretical probability distributions; including the binomial and normal distributions; rules of probability; including conditional probability and expected value; the logic of hypothesis testing; including stating hypotheses; calculating and interpreting p-values; drawing conclusions; and Type I and Type II errors; using confidence intervals to estimate parameters; and proper methods of data collection; including sampling and experimentation. Use of technology; including online applets and the graphing calculator will be prominent in the course. Throughout the course; students will complete investigations that require students to complete the four-step statistical process using athletes of their choice.

Pre-Requisite
Passing grade in MAT100 IGCSE Mathematics I or MAT200 IGCSE Mathematics II

Context for Course
In Arizona; students are now required to take four years of math. Also; in colleges and universities; not only do they expect students to have four (or more) years of high school math; many majors require students to take a course in statistical reasoning. This 21st century skill is a reflection of the increasingly data driven world that we live in. The purpose of this course; Statistical Reasoning in Sports; is to provide students with a class that introduces them to statistical reasoning in a context that is rich with examples likely to spark their interest. And; although the course uses sports as a context; it isn’t primarily about sports—its focus is to teach students about statistics.
Although an Algebra 2 course is recommended as a prerequisite; the only true prerequisites are the ability to think and a willingness to learn—allowing the course to be a fresh start for students who may struggle in the traditional mathematics curriculum. It is also a course for strong math students who are interested in learning about statistics in a different context. You will never hear a student in this class ask “how will this be used in the real world?” From the very first day of class students will be using real data to answer interesting questions.
This sports-based approach is a way to "flavor" the traditional statistics curriculum; extend and deepen learning; and provide real-world applications for core statistical knowledge to better engage students in their learning.

Course goals and major student outcomes
• Students are able to formulate statistical questions and identify statistical claims made by others.
• Students can collect appropriate data to answer statistical questions; including designing experiments and using available data from the internet and other sources.
• Students can use a wide variety of tools to analyze and summarize distributions of data.
• Students understand the role of variability in the data collection process and incorporate this understanding when drawing conclusions about statistical questions.
• Students critically reflect on their own conclusions and conclusions made by others; including the limitations of these conclusions.

Course Objectives
Students will be able to:
• Ask statistical questions and decide which type of data collection procedure is most appropriate in a given situation
• Collect data using online databases in an efficient and effective manner
• Describe the importance of random sampling and the difference between samples and populations
• Design experiments and understand the purpose of control; randomization; replication; and blinding
• Explain how using paired data can provide better results in experiments and observational studies
• Understand when it is appropriate to conclude that changes in one variable cause changes in another variable
• Critically reflect on the limitations of their own conclusions and conclusions made by others
• Summarize distributions of data with appropriate visual displays; including pie charts; bar charts; segmented bar charts; dotplots; histograms; boxplots; and timeplots
• Calculate and interpret summary statistics for numerical data; including the mean; median; range; quartiles; interquartile range; mean absolute deviation and standard deviation.
• Identify unusual values (outliers) in a distribution and understand their effect on summary statistics
• Compare distributions of numerical data; including comparisons of shape; center and spread.
• Use standardized scores to compare athletic performances measured on different scales
• Summarize the relationship between two quantitative variables; using scatterplots; the correlation coefficient; least squares regression lines; and standard deviation of the residuals; and how these measures are affected by unusual values (outliers)
• Explain the principle of least squares
• Describe the concept of regression to the mean
• Use multiple regression to model the relationship between a response variable and several explanatory variables
• Use residual plots to assess the appropriateness of a model
• Use quadratic and exponential functions to model nonlinear associations between two numerical variables
• Use logistic functions to model the relationship between a numerical explanatory variable and a categorical response variable
• Design and conduct simulations to estimate probability distributions; by hand and with technology
• Describe the law of large numbers and its role in simulation
• Explain the concept of independence
• Use the Normal distribution and binomial distribution to estimate probabilities
• Use basic counting rules; including the fundamental counting principle; permutations; and combinations
• Use probability rules; including the addition; multiplication; and complement rules
• Calculate and interpret conditional probabilities
• Calculate and interpret the expected value of a random variable
• Use conditional probability and expected values to evaluate strategies in sports
• Describe the relationship between statistics and parameters in the context of sports
• Explain the concept of sampling variability in the context of sports and its role in the decision making process
• State hypotheses; including a null and alternative hypothesis; about a single proportion; the difference between two proportions; independence of athletic performances; the difference between two means or medians; the mean difference; the difference between two standard deviations; the correlation coefficient; and the slope of a least squares regression line
• Use hands-on methods and technology to simulate the distribution of a single proportion; the difference between two proportions; the longest streak; the number of streaks; the difference between two means or medians; the mean difference; the difference between two standard deviations; the correlation coefficient; and the slope of a least squares regression line
• Estimate and interpret p-values using the results of simulations
• Use p-values to make appropriate conclusions about hypotheses
• Describe a Type I and Type II error in the context of a study and how to reduce the probability that they occur
• Calculate and interpret a confidence interval for a single proportion; a single mean (or mean difference); the difference between two proportions; and the difference between two means
• Use confidence intervals to make decisions

Course Outline:
1: Exploring Categorical Data
Did LeBron James Choke in the Playoffs?
• Distinction between categorical and quantitative variables
• Displays of categorical data; including bar charts; pie charts; and segmented bar charts
• Distinction between parameters and statistics in a sports context
• The law of large numbers
• The use of simulations; by hand and with technology; to investigate variability in athletic performance
• The concept of sampling variability in the context of sports and its role in the decision making process
• How to make conclusions based on the results of simulations; including common errors made in conclusions
• Misleading graphs

2: Comparing Two Proportions
Is There a Home Field Advantage in the NFL?
• How to state hypotheses; including a null and alternative hypothesis; about the difference between two proportions
• How to simulate the distribution of the difference in two proportions
• How to estimate and interpret p-values
• Using p-values to make conclusions about the difference in two proportions
• The vocabulary and principles of experimental design; including explanatory and response variables; treatments; control; and randomization
• The concept of confounding and the types of conclusions that can be drawn from various types of studies; including cause-and-effect conclusions

3: Investigating Independence
Does the Hot Hand Exist in Sports?
• The concept of independence in sports
• Stating hypotheses about independence
• Using different test statistics to measure the hot hand; including simulating the distribution of the test statistics; by hand and with technology
• Statistical significance and significance levels
• Type I and Type II errors; and how to avoid them

4: Exploring Numerical Data
Does the Designated Hitter Increase Offense in Major League Baseball?
• Displays of numerical variables; including dotplots; histograms; and boxplots
• Describing the shape of a distribution
• Measuring the center of a distributions using the mean and the median
• Measuring the spread of a distribution using the range and the interquartile range
• Outliers; how they can be identified; and how they affect measures of center and measures of spread

5: Comparing Two Means or Two Medians
Does the Designated Hitter Increase Offense in Major League Baseball?
• Stating hypotheses about the difference between two means or two medians
• Simulating the distribution of the difference in two means or two medians; by hand and using technology
• Additional concepts in experimental design; including blindness and replication

6: Exploring Paired Data
Can Polyurethane Suits Make You Swim Faster?
• Using paired data to control a source of variability in experiments and observational studies
• The distinction between paired data and unpaired data
• Analyzing paired data using the difference in each pair and the mean difference
• Stating hypotheses about the mean difference
• Simulating the distribution of the mean difference; by hand and using technology

7: Exploring Measures of Variability
Which 7-Iron is More Consistent?
• Using the mean absolute deviation as a measure of consistency/variability
• Using the standard deviation as a measure of consistency/variability
• The influence of outliers on the mean absolute deviation and standard deviation
• Stating hypotheses about the difference of two standard deviations; including the distinction between an athlete’s true standard deviation and the athlete’s observed standard deviation
• Simulating the distribution of the difference in two standard deviations; by hand and with technology

8: Standardized Scores and Normal Distributions
Which Players Should I Draft for My Fantasy League Team?
• Calculating; interpreting and using standardized scores to compare athletic performances from different eras or measured with different units
• Using the 68-95-99.7 rule for approximately Normal distributions
• Using the Normal distribution to model athletic performance
• Using the Normal distribution to estimate probabilities and percentiles

9: Estimating Ability with Confidence Intervals
What is LeBron’s True Ability?
• The logic of confidence intervals; including interpretations of confidence intervals and confidence levels
• Using confidence intervals to make decisions
• Calculating a confidence interval for a single proportion
• Calculating a confidence interval for a single mean or mean difference
• Calculating a confidence interval for the difference between two proportions
• Calculating a confidence interval for the difference between two means

10: Exploring Relationships Between Numerical Variables
Teeing Off: Hit it Long or Hit is Straight?
• Using scatterplots to display the relationship between two quantitative variables
• Describing associations in scatterplots using the characteristics of direction; form; and strength
• Using the correlation coefficient to measure the strength of a linear relationship
• Properties of the correlation coefficient; including how outliers influence the correlation coefficient
• Stating hypotheses about the correlation coefficient; including the distinction between the true correlation and the observed correlation
• Simulating the distribution of the correlation coefficient
• Using time plots and moving averages to display athletic performances over time

11: Using Relationships to Make Predictions
How Can We Build a Better Baseball Team?
• Using equations to make predictions
• Calculating and interpreting residuals
• The concept of least squares
• Calculating and using least-squares regression lines
• Interpreting the slope of a least-squares regression line
• Calculating and interpreting the standard deviation of the residuals
• How outliers influence the equation of a least squares regression line and the standard deviation of the residuals
• Stating hypotheses about the slope of a least-squares regression line; including the distinction between the true slope and the observed slope
• Simulating the distribution of the slope of a least-squares regression line
• Regression to the mean

12: Multiple Regression
Hit it Long or Hit it Straight? Why Not Both?
• The concept of multiple regression
• Using multiple regression models to make predictions
• Using indicator variables in multiple regression models
• Calculating and interpreting the standard deviation of the residuals
• Variable selection

13: Nonlinear Relationships
Will She Make the Shot?
• Using residual plots to assess the appropriateness of a model and to choose between models
• Using quadratic models
• Calculating and interpreting the vertex of the graph of a quadratic model
• Using exponential models
• Interpreting the base and coefficient in an exponential model
• Using logistic models to predict the outcome of a categorical variable

14: Exploring Counting Rules and Probability
How Crazy Was Billy Martin?
• Basic counting rules; including the fundamental counting principal; permutations; and combinations
• Random sampling and the distinction between samples and populations
• The idea of probability; including sample spaces and events
• Basic probability rules and when they are appropriate to use; including the addition; multiplication; and complement rules
• The binomial distribution; including the mean (expected value) of a binomial random variable

15: Conditional Probability; Expected Value; and Strategy in Sports
Should You Go For It on Fourth Down?
• Two-way tables and the general addition rule
• Conditional probability and independence
• Tree diagrams and the general multiplication rule
• Random variables and probability distributions
• Expected values and their interpretations
• Using conditional probabilities and expected values to evaluate strategies in sports

Textbook and Supplementary Materials
• Statistical Reasoning in Sports; by Josh Tabor and Christine Franklin; published by W.H. Freeman.
• Supplementary materials include graphing calculators and Fathom statistical software.

Instructional Methods and/or Strategies
The organization of the content in this course is very different from a traditional statistics course. Instead of taking the first half of the course to build the skills needed to do inference; we will complete the four step statistical process in each chapter. Beginning with the first chapter; we will learn how to ask a statistical question; learn how to collect the appropriate data; learn the skills needed to analyze the data and draw conclusions from the data. In subsequent chapters we will repeat this process; each time with a new focus or type of data.
To make it possible to do inference early in the course; we will be using randomization tests rather than tests using approximations based on a normal model. Although this technique is relatively new in an introductory level statistics course; it is extremely powerful because of its versatility and ability to be easily understood by students while still being theoretically correct. It also enables students to understand the reasoning of hypothesis testing rather than having students memorize a set of algorithms and formulas. For each new type of test; we will begin with hands-on simulations and then follow with technology.
Students will also spend time during each chapter working on projects where they collect data through experimentation or online research. Frequently allowing the students the opportunity to investigate statistical questions of their own choosing will make the material more relevant to the students and also more likely to be remembered.
Finally; technology will play a big part in this day-to-day instruction. Not only will students be using graphing calculators to create graphs and calculate summary statistics; they will be using online applets to run simulations and spend time online doing research and collecting data.

Assessment Methods and/or Tools
• Daily homework
• Daily classwork
• Tests (one per chapter—15 total)
• Individual and Group Investigations (one per chapter—15 total)
• Comprehensive final exam (one per semester)

Alignment with Common Core State Standards for High School Statistics and Probability

Interpreting Categorical and Quantitative Data (S-ID)
Summarize; represent; and interpret data on a single count or measurement variable
Chapters 4-9
Represent data with plots on the real number line (dot plots; histograms; and box plots).

Chapters 4;5;7
Use statistics appropriate to the shape of the data distribution to compare center (median; mean) and spread (interquartile range; standard deviation) of two or more different data sets.

Chapters 4;5;7
Interpret differences in shape; center; and spread in the context of the data sets; accounting for possible effects of extreme data points (outliers).

Chapter 8
Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators; spreadsheets; and tables to estimate areas under the normal curve.

Summarize; represent; and interpret data on two categorical and quantitative variables

Chapter 2;3;15
Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint; marginal; and conditional relative frequencies). Recognize possible associations and trends in the data.

Chapter 10-13
Represent data on two quantitative variables on a scatter plot; and describe how the variables are related.

Chapter 11-13
Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear; quadratic; and exponential models.

Chapter 13
Informally assess the fit of a function by plotting and analyzing residuals.

Chapter 11-13
Fit a linear function for a scatter plot that suggests a linear association.

Interpret linear models

Chapter 11-13
Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.

Chapter 10
Compute (using technology) and interpret the correlation coefficient of a linear fit.

Chapter 1-13
Distinguish between correlation and causation.

Making Inferences and Justifying Conclusions (S-IC)
Understand and evaluate random processes underlying statistical experiments

Chapter 14
Understand statistics as a process for making inferences about population parameters based on a random sample from that population.

Chapter 1-11
Decide if a specified model is consistent with results from a given data-generating process; e.g.; using simulation. For example; a model says a spinning coin falls heads up with probability 0.5. Would a result of 5 tails in a row cause you to question the model?

Make inferences and justify conclusions from sample surveys; experiments; and observational studies

Chapter 2;5;6;7;9;14
Recognize the purposes of and differences among sample surveys; experiments; and observational studies; explain how randomization relates to each.

Chapter 9
Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling.

Chapter 2;5;6;7;9
Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant.

Chapter 1-15
Evaluate reports based on data.

Conditional Probability and the Rules of Probability (S-CP)
Understand independence and conditional probability and use them to interpret data

Chapter 14
Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes; or as unions; intersections; or complements of other events (“or;” “and;” “not”).

Chapter 14
Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities; and use this characterization to determine if they are independent.

Chapter 15
Understand the conditional probability of A given B as P(A and B)/P(B); and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A; and the conditional probability of B given A is the same as the probability of B.

Chapter 2;3;15
Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities.

Chapter 2;3;15
Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations.

Use the rules of probability to compute probabilities of compound events in a uniform probability model

Chapter 15
Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A; and interpret the answer in terms of the model.

Chapter 15
Apply the Addition Rule; P(A or B) = P(A) + P(B) – P(A and B); and interpret the answer in terms of the model.

Chapter 15
Apply the general Multiplication Rule in a uniform probability model; P(A and B) = P(A)P(B|A) = P(B)P(A|B); and interpret the answer in terms of the model.

Chapter 14
Use permutations and combinations to compute probabilities of compound events and solve problems.

Using Probability to Make Decisions (S-MD)
Calculate expected values and use them to solve problems

Chapter 15
Define a random variable for a quantity of interest by assigning a numerical value to each event in a sample space; graph the corresponding probability distribution using the same graphical displays as for data distributions.

Chapter 14;15
Calculate the expected value of a random variable; interpret it as the mean of the probability distribution.

Chapter 14;15
Develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated; find the expected value.

Chapter 15
Develop a probability distribution for a random variable defined for a sample space in which probabilities are assigned empirically; find the expected value.

Use probability to evaluate outcomes of decisions

Chapter 15
Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values.

Chapter 15
Find the expected payoff for a game of chance.

Chapter 15
Evaluate and compare strategies on the basis of expected values.

Chapter 1-3;5-7; 9-11;14;15
Use probabilities to make fair decisions (e.g.; drawing by lots; using a random number generator).

Chapter 15
Analyze decisions and strategies using probability concepts (e.g.; product testing; medical testing; pulling a hockey goalie at the end of a game).

School country

United States

School state

Arizona

School city

Yuma

School / district Address

3150 South Avenue A

School zip code

85364

Requested competency code

Math

Date submitted

05/14/2015

Approved

Yes

Approved competency code

• MTHA
• 4 years of Math

Approved date

01/1/2000

Online / Virtual

No