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## AP®︎/College Statistics

Unit 1: exploring categorical data, unit 2: exploring one-variable quantitative data: displaying and describing, unit 3: exploring one-variable quantitative data: summary statistics, unit 4: exploring one-variable quantitative data: percentiles, z-scores, and the normal distribution, unit 5: exploring two-variable quantitative data, unit 6: collecting data, unit 7: probability, unit 8: random variables and probability distributions, unit 9: sampling distributions, unit 10: inference for categorical data: proportions, unit 11: inference for quantitative data: means, unit 12: inference for categorical data: chi-square, unit 13: inference for quantitative data: slopes, unit 14: prepare for the 2022 ap®︎ statistics exam.

Helping math teachers bring statistics to life

## Probability, Random Variables, And Probability Distributions

Unit 4 Lesson Plans

## AP Exam Review

Chapter 5: Probability

Day 1: Randomness & Probability Day 2: Simulation Day 3: Probability Rules Day 4: General Addition Rule Day 5: Quiz 5.1-5.2 Day 6: Independent & Dependent Events Day 7: Conditional Probability & Tree Diagrams Day 8: Quiz 5.3 Day 9: Chapter 5 Review Day 10: Chapter 5 Test

Chapter 6: Random Variables

Day 1: Discrete Random Variables Day 2: Continuous Random Variables Day 3: Transforming Random Variables Day 4: Combining Random Variables

Day 5: Quiz 6.1-6.2 Day 6: Binomial Distributions Day 1 Day 7: Binomial Distributions Day 2 Day 8: Binomial Distributions Day 3 Day 9: Geometric Distributions Day 10: Quiz 6.3 Day 11: Chapter 6 Review Day 12: Chapter 6 Test

Unit 4 Important Ideas

## AP Statistics Unit 4: Probability, Random Variables, and Probability Distributions

Oct 25, 2022 | AP Statistics

Welcome all! Here, let’s discuss what this unit looks like in my classroom. I’m hoping this blog post will help you start planning out your own probability unit in your AP Statistics classrooms. Keep reading to see how I balance pacing, summary days, and hands-on learning in Unit 4.

## Unit 4 – Almost 4 weeks?

The AP © Stat CED recommends spending 18 – 20 days on this unit. I laid out everything in the pacing guide so that this unit is completed in 19 days (check out my FREE pacing guide at the end of the blog!). We know that timing is always tight right as we get to Unit 4, so I do have some suggestions under “Questions on Timing” below that could help you!

This year, I rearranged my notes so that I’m doing smaller lessons (25-40 minutes) and then giving a homework assignment for students to practice material. Here is how I layout my notes:

Notes 1 – Basic Probability and Simulations

Notes 2 – The Addition Rule

Notes 3 – Venn Diagrams, Unions, and Intersections

Notes 4 – The Multiplication Rule and Conditional Probability

Notes 5 – Discrete and Continuous Random Variables

Notes 6 – Combining Random Variables

Notes 7 – The Binomial Distribution

Notes 8 – The Geometric Distribution

As always, there are videos of me teaching these materials that go along with each set of notes. I’ve found these videos to be SO HELPFUL for students who miss class, or if I am ever absent a few days during the unit.

I still really enjoy giving out the note packets at the beginning of the unit for the students. If they are ever absent, if I am, or if our copier is just broken (let’s be honest, that happens ALL. THE. TIME.), it is nice to still have all the notes printed and not have to worry about them.

## Summary Days

By now, you’ve seen that I love a good review day. My students have always said that they love having a day where I summarize the major points in the unit, and then give them time to work through problems and ask their questions.

I have two teacher-lead review days in this unit. There are slides for you to flip through that show the answers to a student handout summarizing the main points. My students LOVE these days, but I know some teachers prefer to have the students be more hands-on these days instead. It is all about what you feel comfortable doing with YOUR students. The third review day (Unit 4 test review day), I do not have any teacher-lead slides. We usually use this day to go over most missed problems from the random variables quiz, and then I give them the rest of the day to start working on the test review and ask questions.

## Question on Timing

When I plan for the year, I always want to get through Unit 4 by the time semester 1 exams roll around. I try to finish Unit 4 with enough time for a couple of days of cumulative review before the semester 1 final.

I’ve been pretty good at doing this the past few years. Sometimes, I have to cut out an activity or two in Unit 4 to make sure I make it, but I usually do. This was not the case at the beginning of my teaching journey!

Before the CED was published, I was lucky to get through conditional probability by the end of the year. All my random variable material was saved for second semester. Now doing this was not impossible, but I barely had time to teach Unit 9 (we are talking, a single class period to go over Unit 9), and I had only about 2 weeks for review, but it was possible!

It also depends on when your school year starts too. We start mid-August in my area, so finishing Unit 4 by winter break is doable. If you don’t start until mid-September, this might be more of a challenge. If you are stressed about time in Unit 4, here are some things that you can do to make sure you get through the material comfortably:

• Make the two quizzes take-home (or get rid of them all together) – this will free up two days for you!
• Make the quizzes shorter, so that you use the review slides for half the period and the shorter quiz for the second half of the period
• The first activity I would cut out would be the binomial distribution of blue (an M&M activity). While I love getting students hands-on with the binomial distribution, this is not necessary to complete the content, but instead is there to help reinforce it. Don’t feel guilty for skipping it.
• You might not also need a full day for the Unit 4 project. I like taking a day to introduce it and then giving students time to work on it. If you are pressed for time, you do not have to use this day for a project.

## Probability Activities

When I started teaching probability, I really struggled with the “hands-on” part of learning probability. It was hard for me to step away from the lecture-practice-assess model, since that was my comfort zone (especially with probability, which I was originally so uncomfortable with!).

I finally have reached a point where I feel really good about the activities my students are doing in this Unit. It breaks up the lecture-practice-assess atmosphere and lets them work with each other on the topics we talk about in class. Here is a brief description of each activity and when I use them:

Introduction to Probability Activity

• This is an activity I do before we start notes for the Unit. I introduce a game called “beat the teacher” and you run through a simulation with the students to see if it is a fair game to play. Students are introduced to the basic probability formula, as well as empirical and theoretical probability terms.

Unit 4 Circuit

• This is a review worksheet in a “circuit style” review. Students work through 14 problems on basic probability rules and conditional probability.

Simulations Activity

• This activity allows students to practice the four-step process while performing 3 simulations.

Simulations and the Binomial

• Students explore conducting a simulation to answer a question and are met with the limitations of conducting simulations. The solution? The Binomial Random Variable!

The Binomial Distribution of Blue Activity

• Students get hands-on with M&Ms to compare the theoretical probabilities obtained with the binomial formulas to the empirical probabilities obtained in a bag of M&Ms.

• Students create 3 games of chance to practice developing a probability distribution: one where the player wins, the house wins, and one where the game breaks even. (Example report and rubric included)

Introduction to the Binomial

• Students will take a quiz in Chinese (to simulate randomly guessing T/F on a quiz), find the probabilities of getting the questions correct, and then see how we can fit a probability distribution (aka, the binomial) to the data.

I also have two additional lessons in my projects and activities package. These lessons are ones I teach in my regular-level statistics course and ones that I use to teach in AP Statistics before the CED redesign in 2020. The content is not included on the AP exam anymore, but they are very closely related to what we are learning about in class:

Permutations and Combinations Lesson

• This is an additional lesson you can use to teach the counting principle, combinations, and permutations. There is a set of notes and a homework assignment.

The Poisson Distribution Lesson

• This is an additional lesson you can use to teach about the Poisson Distribution. There is a set of notes and a homework assignment.

If you have made it this far in my blog post, thank you! I really hope that some of the things I have gone over can help you in your classroom. Check out my freebies for this unit below and as always, feel free to email me any time with any questions.

• Unit 4 Pacing and Objectives
• Binomial vs Geometric Random Variables Summary Sheet

• AP Statistics Syllabus
• AP Statistics PowerPoints

• Homework Unit 1 – 1st part
• Homework Unit 1 – 2nd part
• Homework Unit 1 – 3rd part

• Unit 2 Homework – Part 1
• Unit 2 Homework – Part 2
• Unit 2 Homework – Part 3
• Unit 2 Homework – Part 4

• Chapter 5, Section 1
• Chapter 5, Section 2
• Chapter 5, Section 3

• TRM-Section 8.1 Full Solutions
• TRM-Section 8.2 Full Solutions
• TRM-Section 8.3 Full Solutions

• Homework Unit 6 – 2019-2020

• HomeLogic Log-in
• Mr. Gallo's Website
• NHHS Moodle
• NHV District Page
• Wordpress Log-in

The relative frequency shows the proportion of data points that have each value. The frequency tells the number of data points that have each value.

Answers will vary. One possible histogram is shown below.

Find the midpoint for each class. These will be graphed on the x -axis. The frequency values will be graphed on the y -axis values.

• The 40 th percentile is 37 years.
• The 78 th percentile is 70 years.

Jesse graduated 37 th out of a class of 180 students. There are 180 – 37 = 143 students ranked below Jesse. There is one rank of 37.

x = 143 and y = 1. x + .5 y n x + .5 y n (100) = 143 + .5 ( 1 ) 180 143 + .5 ( 1 ) 180 (100) = 79.72. Jesse’s rank of 37 puts him at the 80 th percentile.

• For runners in a race, it is more desirable to have a high percentile for speed. A high percentile means a higher speed, which is faster.
• 40 percent of runners ran at speeds of 7.5 miles per hour or less (slower), and 60 percent of runners ran at speeds of 7.5 miles per hour or more (faster).

When waiting in line at the DMV, the 85 th percentile would be a long wait time compared to the other people waiting. 85 percent of people had shorter wait times than Mina. In this context, Mina would prefer a wait time corresponding to a lower percentile. 85 percent of people at the DMV waited 32 minutes or less. 15 percent of people at the DMV waited 32 minutes or longer.

The manufacturer and the consumer would be upset. This is a large repair cost for the damages, compared to the other cars in the sample. INTERPRETATION: 90 percent of the crash-tested cars had damage repair costs of \$1,700 or less; only 10 percent had damage repair costs of \$1,700 or more.

You can afford 34 percent of houses. 66 percent of the houses are too expensive for your budget. INTERPRETATION: 34 percent of houses cost \$240,000 or less; 66 percent of houses cost \$240,000 or more.

More than 25 percent of salespersons sell four cars in a typical week. You can see this concentration in the box plot because the first quartile is equal to the median. The top 25 percent and the bottom 25 percent are spread out evenly; the whiskers have the same length.

Mean: 16 + 17 + 19 + 20 + 20 + 21 + 23 + 24 + 25 + 25 + 25 + 26 + 26 + 27 + 27 + 27 + 28 + 29 + 30 + 32 + 33 + 33 + 34 + 35 + 37 + 39 + 40 = 738;

738 27 738 27 = 27.33

The most frequent lengths are 25 and 27, which occur three times. Mode = 25, 27

The data are symmetrical. The median is 3, and the mean is 2.85. They are close, and the mode lies close to the middle of the data, so the data are symmetrical.

The data are skewed right. The median is 87.5, and the mean is 88.2. Even though they are close, the mode lies to the left of the middle of the data, and there are many more instances of 87 than any other number, so the data are skewed right.

When the data are symmetrical, the mean and median are close or the same.

The distribution is skewed right because it looks pulled out to the right.

The mean is 4.1 and is slightly greater than the median, which is 4.

The mode and the median are the same. In this case, both 5.

The distribution is skewed left because it looks pulled out to the left.

Both the mean and the median are 6.

The mode is 12, the median is 13.5, and the mean is 15.1. The mean is the largest.

The mean tends to reflect skewing the most because it is affected the most by outliers.

sampling variability

induced variability

measurement variability

natural variability

For Fredo: z = .158  –  .166 .012 .158  –  .166 .012 = –0.67.

For Karl: z = .177  –  .189 .015 .177  –  .189 .015 = –.8.

Fredo’s z score of –.67 is higher than Karl’s z score of –.8. For batting average, higher values are better, so Fredo has a better batting average compared to his team.

• s x = ∑ f m 2 n − x ¯ 2 = 193,157.45 30 − 79.5 2 = 10.88 s x = ∑ f m 2 n − x ¯ 2 = 193,157.45 30 − 79.5 2 = 10.88
• s x = ∑ f m 2 n − x ¯ 2 = 380,945.3 101 − 60.94 2 = 7.62 s x = ∑ f m 2 n − x ¯ 2 = 380,945.3 101 − 60.94 2 = 7.62
• s x = ∑ f m 2 n − x ¯ 2 = 440,051.5 86 − 70.66 2 = 11.14 s x = ∑ f m 2 n − x ¯ 2 = 440,051.5 86 − 70.66 2 = 11.14
• Number the entries in the table 1–51 (includes Washington, DC; numbered vertically)
• Arrow over to PRB
• Press 5:randInt(
• Enter 51,1,8)

Eight numbers are generated (use the right arrow key to scroll through the numbers). The numbers correspond to the numbered states (for this example: {47 21 9 23 51 13 25 4}. If any numbers are repeated, generate a different number by using 5:randInt(51,1)). Here, the states (and Washington DC) are {Arkansas, Washington DC, Idaho, Maryland, Michigan, Mississippi, Virginia, Wyoming}.

Corresponding percents are {30.1, 22.2, 26.5, 27.1, 30.9, 34.0, 26.0, 25.1}.

• See Table 2.89 and Table 2.90 .
• Both graphs have a single peak.
• Both graphs use class intervals with width equal to \$50
• The couples graph has a class interval with no values
• It takes almost twice as many class intervals to display the data for couples
• Answers may vary. Possible answers include the following. The graphs are more similar than different because the overall patterns for the graphs are the same.
• Check student's solution.
• Both graphs have a single peak
• Both graphs display six class intervals
• Both graphs show the same general pattern
• Answers may vary. Possible answers include the following. Although the width of the class intervals for couples is double that of the class intervals for singles, the graphs are more similar than they are different.
• Answers may vary. Possible answers include the following. You are able to compare the graphs interval by interval. It is easier to compare the overall patterns with the new scale on the couples graph. Because a couple represents two individuals, the new scale leads to a more accurate comparison.
• Answers may vary. Possible answers include the following. Based on the histograms, it seems that spending does not vary much from singles to individuals who are part of a couple. The overall patterns are the same. The range of spending for couples is approximately double the range for individuals.

• 1 – (.02+.09+.19+.26+.18+.17+.02+.01) = .06
• .19+.26+.18 = .63
• Check student’s solution.

40 th percentile will fall between 30,000 and 40,000

80 th percentile will fall between 50,000 and 75,000

• more children; the left whisker shows that 25 percent of the population are children 17 and younger; the right whisker shows that 25 percent of the population are adults 50 and older, so adults 65 and over represent less than 25 percent
• 62.4 percent
• Answers will vary. Possible answer: State University conducted a survey to see how involved its students are in community service. The box plot shows the number of community service hours logged by participants over the past year.
• Because the first and second quartiles are close, the data in this quarter is very similar. There is not much variation in the values. The data in the third quarter is much more variable, or spread out. This is clear because the second quartile is so far away from the third quartile.
• Each box plot is spread out more in the greater values. Each plot is skewed to the right, so the ages of the top 50 percent of buyers are more variable than the ages of the lower 50 percent.
• The black sports car is most likely to have an outlier. It has the longest whisker.
• Comparing the median ages, younger people tend to buy the black sports car, while older people tend to buy the white sports car. However, this is not a rule, because there is so much variability in each data set.
• The second quarter has the smallest spread. There seems to be only a three-year difference between the first quartile and the median.
• The third quarter has the largest spread. There seems to be approximately a 14-year difference between the median and the third quartile.
• IQR ~ 17 years
• There is not enough information to tell. Each interval lies within a quarter, so we cannot tell exactly where the data in that quarter is are concentrated.
• The interval from 31 to 35 years has the fewest data values. Twenty-five percent of the values fall in the interval 38 to 41, and 25 percent fall between 41 and 64. Since 25 percent of values fall between 31 and 38, we know that fewer than 25 percent fall between 31 and 35.

the mean percentage, x ¯ = 1,328.65 50 = 26.75 x ¯ = 1,328.65 50 = 26.75

The median value is the middle value in the ordered list of data values. The median value of a set of 11 will be the sixth number in order. Six years will have totals at or below the median.

• mean = 1,809.3
• median = 1,812.5
• standard deviation = 151.2
• first quartile = 1,690
• third quartile = 1,935

Hint: think about the number of years covered by each time period and what happened to higher education during those periods.

For pianos, the cost of the piano is .4 standard deviations BELOW the mean. For guitars, the cost of the guitar is 0.25 standard deviations ABOVE the mean. For drums, the cost of the drum set is 1.0 standard deviations BELOW the mean. Of the three, the drums cost the lowest in comparison to the cost of other instruments of the same type. The guitar costs the most in comparison to the cost of other instruments of the same type.

• x ¯ = 23.32 x ¯ = 23.32
• Using the TI 83/84, we obtain a standard deviation of: s x = 12.95. s x = 12.95.
• The obesity rate of the United States is 10.58 percent higher than the average obesity rate.
• Since the standard deviation is 12.95, we see that 23.32 + 12.95 = 36.27 is the disease percentage that is one standard deviation from the mean. The U.S. disease rate is slightly less than one standard deviation from the mean. Therefore, we can assume that the United States, although 34 percent have the disease, does not have an unusually high percentage of people with the disease.
• For graph, check student's solution.
• 49.7 percent of the community is under the age of 35
• Based on the information in the table, graph (a) most closely represents the data.
• 174, 177, 178, 184, 185, 185, 185, 185, 188, 190, 200, 205, 205, 206, 210, 210, 210, 212, 212, 215, 215, 220, 223, 228, 230, 232, 241, 241, 242, 245, 247, 250, 250, 259, 260, 260, 265, 265, 270, 272, 273, 275, 276, 278, 280, 280, 285, 285, 286, 290, 290, 295, 302
• 205.5, 272.5
• .84 standard deviations below the mean

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Want to cite, share, or modify this book? This book uses the Creative Commons Attribution License and you must attribute Texas Education Agency (TEA). The original material is available at: https://www.texasgateway.org/book/tea-statistics . Changes were made to the original material, including updates to art, structure, and other content updates.

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AP STATISTIC (PDF) WORKSHEETS ( W ), HANDOUTS ( H ), APPLETS ( A ), VIDEOS ( V ) & COMMENTARIES ( C ) FROM AP TEACHERS THROUGH AP CENTRAL

( Many video lessons are available through www.khanacademy.org/math/ap-statistics and on the homepage of AP Classroom through My AP )

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AP®︎/College Statistics 14 units · 137 skills. Unit 1 Exploring categorical data. Unit 2 Exploring one-variable quantitative data: Displaying and describing. Unit 3 Exploring one-variable quantitative data: Summary statistics. Unit 4 Exploring one-variable quantitative data: Percentiles, z-scores, and the normal distribution.

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