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The Practice of Statistics for AP

Daren S. Starnes, Daniel S. Yates, David S. Moore

Chapter 2

Modeling Distributions of Data - all with Video Answers

Educators

Section 1

Describing Location in a Distribution

02:32

Problem 1

Shoes How many pairs of shoes do students have? Do girls have more shoes than boys? Here are data from a random sample of 20 female and 20 male students at a large high school: $$
\begin{array}{llrrrrrrrrr}
\hline \text { Female: } & 50 & 26 & 26 & 31 & 57 & 19 & 24 & 22 & 23 & 38 \\
& 13 & 50 & 13 & 34 & 23 & 30 & 49 & 13 & 15 & 51 \\
\text { Male: } & 14 & 7 & 6 & 5 & 12 & 38 & 8 & 7 & 10 & 10 \\
& 10 & 11 & 4 & 5 & 22 & 7 & 5 & 10 & 35 & 7 \\
\hline
\end{array}
$$
(a) Find and interpret the percentile in the female distribution for the girl with 22 pairs of shoes.
(b) Find and interpret the percentile in the male distribution for the boy with 22 pairs of shoes.
(c) Who is more unusual: the girl with 22 pairs of shoes or the boy with 22 pairs of shoes? Explain.

John Long
John Long
Numerade Educator
03:01

Problem 2

Old folks Here is a stemplot of the percents of residents aged 65 and older in the 50 states:
(a) Find and interpret the percentile for Colorado, where $10.1 \%$ of the residents are aged 65 and older.
(b) Find and interpret the percentile for Rhode Island, where $13.9 \%$ of the residents are aged 65 and older.
(c) Which of these two states is more unusual? Explain.

Bryan Meares
Bryan Meares
Numerade Educator
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Problem 3

Math test Josh just got the results of the statewide Algebra 2 test: his score is at the 60 th percentile. When Josh gets home, he tells his parents that he got 60 percent of the questions correct on the state test. Explain what's wrong with Josh's interpretation.

Donna Densmore
Donna Densmore
Numerade Educator
01:03

Problem 4

Blood pressure Larry came home very excited after a visit to his doctor. He announced proudly to his wife, "My doctor says my blood pressure is at the 90 th percentile among men like me. That means I'm better off than about $90 \%$ of similar men." How should his wife, who is a statistician, respond to Larry's statement?

R M
R M
Numerade Educator
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Problem 5

Growth charts We used an online growth chart to find percentiles for the height and weight of a l6-yearold girl who is 66 inches tall and weighs 118 pounds. According to the chart, this girl is at the 48 th percentile for weight and the 78 th percentile for height. Explain what these values mean in plain English.

Donna Densmore
Donna Densmore
Numerade Educator
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Problem 6

Run fast Peter is a star runner on the track team. In the league championship meet, Peter records a time that would fall at the 80 th percentile of all his race times that season. But his performance places him at the 50 th percentile in the league championship meet. Explain how this is possible. (Remember that lower times are better in this case!)

Rashmi Sinha
Rashmi Sinha
Numerade Educator
02:05

Problem 7

Each point gives the value of the variable being measured and the corresponding percentile for one individual in the data set.
Text me The percentile plot below shows the distribution of text messages sent and received in a two-day period by a random sample of 16 females from a large high school.
(a) Describe the student represented by the highlighted point.
(b) Use the graph to estimate the median number of texts. Explain your method.

John Long
John Long
Numerade Educator
02:19

Problem 8

Each point gives the value of the variable being measured and the corresponding percentile for one individual in the data set.
Foreign-born residents The following percentile plot shows the distribution of the percent of foreign-born residents in the 50 states.
(a) The highlighted point is for Maryland. Describe what the graph tells you about this state.
Use the graph to estimate the 30 th percentile of the distribution. Explain your method.

John Long
John Long
Numerade Educator
04:56

Problem 9

Shopping spree The figure below is a cumulative relative frequency graph of the amount spent by 50 consecutive grocery shoppers at a store.
(a) Estimate the interquartile range of this distribution. Show your method.
(b) What is the percentile for the shopper who spent $\$ 19.50 ?$
(c) Draw the histogram that corresponds to this graph.

John Long
John Long
Numerade Educator
03:26

Problem 10

Light it up! The graph below is a cumulative relative frequency graph showing the lifetimes (in hours) of 200 lamps. ${ }^{4}$
(a) Estimate the 60 th percentile of this distribution. Show your method.
(b) What is the percentile for a lamp that lasted 900 hours?
(c) Draw a histogram that corresponds to this graph.

John Long
John Long
Numerade Educator
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Problem 11

Eleanor scores 680 on the SAT Mathematics test. The distribution of SAT scores is symmetric and single-peaked, with mean 500 and standard deviation 100 . Gerald takes the American College Testing (ACT) Mathematics test and scores 27. ACT scores also follow a symmetric, single-peaked distribution-but with mean 18 and standard deviation $6 .$ Find the standardized scores for both students. Assuming that both tests measure the same kind of ability, who has the higher score?

Donna Densmore
Donna Densmore
Numerade Educator
04:52

Problem 12

Comparing batting averages Three landmarks of baseball achievement are Ty Cobb's batting average of 0.420 in $1911,$ Ted Williams's 0.406 in $1941,$ and George Brett's 0.390 in $1980 .$ These batting averages cannot be compared directly because the distribution of major league batting averages has changed over the years. The distributions are quite symmetric, except for outliers such as Cobb, Williams, and Brett. While the mean batting average has been held roughly constant by rule changes and the balance between hitting and pitching, the standard deviation has dropped over time. Here are the facts: $$
\begin{array}{clc}
\hline \text { Decade } & \text { Mean } & \text { Standard deviation } \\
1910 \mathrm{~s} & 0.266 & 0.0371 \\
1940 \mathrm{~s} & 0.267 & 0.0326 \\
1970 \mathrm{~s} & 0.261 & 0.0317 \\
\hline
\end{array}
$$
Find the standardized scores for Cobb, Williams, and Brett. Who was the best hitter?$^5$

John Long
John Long
Numerade Educator
03:12

Problem 13

Measuring bone density Individuals with low bone density have a high risk of broken bones (fractures). Physicians who are concerned about low bone density (osteoporosis) in patients can refer them for specialized testing. Currently, the most common method for testing bone density is dual-energy X-ray absorptiometry (DFXA). A patient who undergoes a DFXA test usually gets bone density results in grams per square centimeter $\left(\mathrm{g} / \mathrm{cm}^{2}\right)$ and in standardized units.
Judy, who is 25 years old, has her bone density measured using DEXA. Her results indicate a bone density in the hip of $948 \mathrm{~g} / \mathrm{cm}^{2}$ and a standardized score of $z=-1.45 .$ In the reference population of 25-year-old women like Judy, the mean bone density in the hip is $956 \mathrm{~g} / \mathrm{cm}^{2} .$

John Long
John Long
Numerade Educator
04:39

Problem 14

Comparing bone density Refer to the previous exercise. One of Judy's friends, Mary, has the bone density in her hip measured using DEXA. Mary is 35 years old. Her bone density is also reported as $948 \mathrm{~g} / \mathrm{cm}^{2}$, but her standardized score is $z=0.50 .$ The mean bone density in the hip for the reference population of 35 -year-old women is 944 grams $/ \mathrm{cm}^{2}$.
(a) Whose bones are healthier - Judy's or Mary's? Justify
your answer.
(b) Calculate the standard deviation of the bone density in Mary's reference population. How does this compare with your answer to Exercise $13(\mathrm{~b})$ ? Are you surprised?

John Long
John Long
Numerade Educator
03:00

Problem 15

Refer to the dotplot and summary statistics of salaries for players on the World Champion 2008 Philadelphia Phillies baseball tedm.
Baseball salaries Brad Lidge played a crucial role as the Phillies' "closer," pitching the end of many games throughout the season. Lidge's salary for the 2008 season was $\$ 6,350,000$.
(a) Find the percentile corresponding to Lidge's salary. Explain what this value means.
(b) Find the $z$ -score corresponding to Lidge's salary. Explain what this value means.

John Long
John Long
Numerade Educator
02:55

Problem 16

Refer to the dotplot and summary statistics of salaries for players on the World Champion 2008 Philadelphia Phillies baseball tedm.
Baseball salaries Did Ryan Madson, who was paid $\$ 1,400,000,$ have a high salary or a low salary compared with the rest of the team? Justify your answer by calculating and interpreting Madson's percentile and $z$ -score.

John Long
John Long
Numerade Educator
01:40

Problem 17

The scores on Ms. Martin's statistics quiz had a mean of 12 and a standard deviation of $3 .$ Ms. Martin wants to transform the scores to have a mean of 75 and a standard deviation of $12 .$ What transformations should she apply to each test score? Explain.

John Long
John Long
Numerade Educator
01:53

Problem 18

Mr. Olsen uses an unusual grading system in his class. After each test, he transforms the scores to have a mean of 0 and a standard deviation of $1 . \mathrm{Mr}$. Olsen then assigns a grade to each student based on the transformed score. On his most recent test, the class's scores had a mean of 68 and a standard deviation of
15. What transformations should he apply to each test score? Explain.

John Long
John Long
Numerade Educator
03:19

Problem 19

Tall or short? Mr. Walker measures the heights (in inches) of the students in one of his classes. He uses a computer to calculate the following numerical summaries: $$
\begin{array}{ccccccc}
\hline \text { Mean } & \text { Std. dev. } & \text { Min } & a_{1} & \text { Med } & a_{3} & \text { Max } \\
69.188 & 3.20 & 61.5 & 67.75 & 69.5 & 71 & 74.5 \\
\hline
\end{array}
$$
Next, Mr. Walker has his entire class stand on their chairs, which are 18 inches off the ground. Then he measures the distance from the top of each student's head to the floor.
(a) Find the mean and median of these measurements. Show your work.
(b) Find the standard deviation and $I Q R$ of these measurements. Show your work.

John Long
John Long
Numerade Educator
03:34

Problem 20

Teacher raises $A$ school system employs teachers at salaries between $\$ 28,000$ and $\$ 60,000$. The teachers" union and the school board are negotiating the form of next year's increase in the salary schedule.
(a) If every teacher is given a flat $\$ 1000$ raise, what will this do to the mean salary? To the median salary? Explain your answers.
(b) What would a flat $\$ 1000$ raise do to the extremes and quartiles of the salary distribution? To the standard deviation of teachers' salaries? Explain your answers.

John Long
John Long
Numerade Educator
03:21

Problem 21

Tall or short? Refer to Exercise 19. Mr. Walker converts his students' original heights from inches to feet.
(a) Find the mean and median of the students' heights in feet. Show your work.
(b) Find the standard deviation and $I Q R$ of the students" heights in feet. Show your work.

John Long
John Long
Numerade Educator
02:44

Problem 22

Teacher raises Refer to Exercise $20 .$ If each teacher receives a $5 \%$ raise instead of a flat $\$ 1000$ raise, the amount of the raise will vary from $\$ 1400$ to $\$ 3000$, depending on the present salary.
(a) What will this do to the mean salary? To the median salary? Explain your answers.
(b) Will a $5 \%$ raise increase the $I Q R ?$ Will it increase the standard deviation? Explain your answers.

John Long
John Long
Numerade Educator
02:07

Problem 23

Cool pool? Coach Ferguson uses a thermometer to measure the temperature (in degrees Celsius) at 20 different locations in the school swimming pool. An analysis of the data yields a mean of $25^{\circ} \mathrm{C}$ and a standard deviation of $2^{\circ} \mathrm{C}$. Find the mean and standard deviation of the temperature readings in degrees Fahrenheit (recall that ${ }^{\circ} \mathrm{F}=(9 / 5)^{\circ} \mathrm{C}+32$ ).

R M
R M
Numerade Educator
02:11

Problem 24

Measure up Clarence measures the diameter of each tennis ball in a bag with a standard ruler. Unfortunately, he uses the ruler incorrectly so that each of his measurements is 0.2 inches too large. Clarence's data had a mean of 3.2 inches and a standard deviation of 0.1 inches. Find the mean and standard deviation of the corrected measurements in centimeters (recall that 1 inch $=2.54 \mathrm{~cm}$ ).

John Long
John Long
Numerade Educator
01:21

Problem 25

Select the best answer
Jorge's score on Fixam 1 in his statistics class was at the 64 th percentile of the scores for all students. His score falls
(a) between the minimum and the first quartile.
(b) between the first quartile and the median.
(c) between the median and the third quartile.
(d) between the third quartile and the maximum.
(e) at the mean score for all students.

John Long
John Long
Numerade Educator
05:51

Problem 26

Select the best answer
When Sam goes to a restaurant, he always tips the server $\$ 2$ plus $10 \%$ of the cost of the meal. If Sam's distribution of meal costs has a mean of $\$ 9$ and a standard deviation of $\$ 3,$ what are the mean and standard deviation of the distribution of his tips?
(a) $\$ 2.90, \$ 0.30$
(b) $\$ 2.90, \$ 2.30$
(c) $\$ 9.00, \$ 3.00$
(d) $\$ 11.00, \$ 2.00$
(e) $\$ 2.00, \$ 0.90$

Sanchit Jain
Sanchit Jain
Numerade Educator
02:23

Problem 27

Scores on the $A C$ T college entrance exam follow a bell-shaped distribution with mean 18 and standard deviation 6 . Wayne's standardized score on the ACT was $-0.5 .$ What was Wayne's actual ACT score?
(a) 5.5
(b) 12
(c) 15
(d) 17.5
(e) 21

Sanchit Jain
Sanchit Jain
Numerade Educator
02:03

Problem 28

George has an average bowling score of 180 and bowls in a league where the average for all bowlers is 150 and the standard deviation is $20 .$ Bill has an average bowling score of 190 and bowls in a league where the average is 160 and the standard deviation is $15 .$ Who ranks higher in his own league, George or Bill?
(a) Bill, because his 190 is higher than George's 180 .
(b) Bill, because his standardized score is higher than George's.
(c) Bill and George have the same rank in their leagues, because both are 30 pins above the mean.
(d) George, because his standardized score is higher than Bill's.
(e) George, because the standard deviation of bowling scores is higher in his league.

John Long
John Long
Numerade Educator
01:00

Problem 29

Refer to the following setting. The number of absences during the fall semester was recorded for each student in a large elementary school. The distribution of absences is displayed in the following cumulative relative frequency graph.
What is the interquartile range $(I Q R)$ for the distribution of absences?
(a) 1
(b) 2
(c) 3
(d) 5
(e) 14

John Long
John Long
Numerade Educator
00:57

Problem 30

Refer to the following setting. The number of absences during the fall semester was recorded for each student in a large elementary school. The distribution of absences is displayed in the following cumulative relative frequency graph.
If the distribution of absences was displayed in a histogram, what would be the best description of the histogram's shape?
(a) Symmetric
(b) Uniform
(c) Skewed left
(d) Skewed right
(e) Cannot be determined

John Long
John Long
Numerade Educator
02:11

Problem 31

Refer to the following setting. We used CensusAtSchool's Random Data Selector to choose a sample of 50 Canadian students who completed a survey in a recent year.
Travel time (1.2) The dotplot below displays data on students' responses to the question "How long does it usually take you to travel to school?" Describe the shape, center, and spread of the distribution. Are there any outliers?

John Long
John Long
Numerade Educator
03:00

Problem 32

Refer to the following setting. We used CensusAtSchool's Random Data Selector to choose a sample of 50 Canadian students who completed a survey in a recent year.
Lefties (1.1) Students were asked, "Are you righthanded, left-handed, or ambidextrous?" The responses are shown below $(\mathrm{R}=$ right-handed; $\mathrm{L}=$ left-handed; $\mathrm{A}=$ ambidextrous $)$
(a) Make an appropriate graph to display these data.
(b) Over 10,000 Canadian high school students took the CensusAtSchool survey that year. What percent of this population would you estimate is left-handed? Justify your answer.

John Long
John Long
Numerade Educator