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LESSON 2.1 • Describing Location in a Distribution 95

x −15.62
d 14. Meaning of the Dow The Dow Jones Industrial
d d d (b) 2.34 = gives x =19.90.
d d d d d d d d
d d d d d d d d d d d d d Average (DJIA) is a commonly used index of the 1.83
60 62 64 66 68 70 72 74 overall strength of the U.S. stock market. In 2019, Florida has 19.9% of residents aged 65
Height (in.) the mean daily change in the DJIA for the days that
the stock markets were open was 20.94 points with or older.
(a) Find the percentile for Lynette, the student who is a standard deviation of 206.77 points.
−
65 inches tall. (a) Find and interpret the standardized score (z-score) 14. (a) z = −473.39 20.94 =−2.39. Lesson 2.1
(b) Asher’s height is at the 88th percentile of the distri- for the change in the DJIA on May 7, 2019, which 206.77
bution. Interpret this value. How tall is Asher? was 473.39− points. The change in the DJIA on May 7 was
9. A boy and his shoes How many pairs of shoes does (b) The standardized score for November 20, 2019, was 2.39 standard deviation below the mean
a typical teenage boy own? To find out, a group of z =− 0.65. Find the change in the DJIA for that date.
statistics students surveyed a random sample of 20 15. SAT versus ACT During her senior year, Courtney change of 20.94 points.
male students from their large high school. Then pg 93 took both the SAT and ACT. She scored 680 on the x −20.94
they recorded the number of pairs of shoes that SAT math test and 27 on the ACT math test. Scores (b) −0.65 = gives x = –113.46.
each boy owned. Here are the data. on the math section of the SAT vary from 200 to 206.77
(C) 2021 BFW Publishers -- for review purposes only.
800, with a mean of 528 and standard deviation of On November 20, 2019 the DJIA went
14 7 6 5 12 38 8 7 10 10 117. Scores on the math section of the ACT vary
10 11 4 5 22 7 5 10 35 7 from 1 to 36, with a mean of 20.5 and a standard down 113.46 points.
4
deviation of 5.5. Calculate Courtney’s standardized
(a) Martin is the student who reported owning 22 score on each test. Which of her two test scores was 680 −528
pairs of shoes. Find Martin’s percentile. better, relatively speaking? Explain your reasoning. 15. SAT: z = 117 =1.30
(b) Luis is at the 25th percentile of the distribution. 16. Generational GPA Rebecca and her father both
How many pairs of shoes does Luis own? graduated from the same high school. When her 27 −20.5
10. Unlocked for sale The “sold” listings on a popular father looked at Rebecca’s transcript, he noticed that ACT: z = 5.5 =1.18
auction website included 20 sales of used “unlocked” her high school GPA (4.2) was higher than his high
phones of one popular model. Here are the sales school GPA (3.9). After letting Rebecca gloat for a Courtney scored better on the SAT
prices. minute, he pointed out that there were no weighted relative to her peers because her z-score
grades when he went to school. To settle their argu-
450 415 495 300 325 430 370 ment, they called the registrar at the school and got on the SAT z =( 1.30) was greater than
400 325 400 235 330 304 415 information about the distribution of GPA in each her z-score on the ACT z =( 1.18).
of their graduation years. When the father gradu-
355 405 449 355 425 299 ated, the mean GPA was 2.8 with a standard devia-
.− .28
tion of 0.6. When Rebecca graduated, the mean GPA 16. Rebecca’s father: z = 39 =.183
(a) Find the percentile of the phone that sold for $330. was 3.2 with a standard deviation of 0.7. Who had . 06
(b) What was the sales price of the phone that was at the better GPA, relatively speaking? Explain your
.− .32
the 75th percentile? reasoning. 42
11. The Nationals play During the 2019 season, the 17. Biles by miles! Simone Biles won the gold medal Rebecca: z = . 07 =.143
pg 92 mean number of wins for Major League Base- in women’s artistic gymnastics at the 2019 World
ball teams was 81 with a standard deviation of Championships. Her overall score in the all-around Rebecca’s father had a better GPA relative
15.9 wins. Find the standardized score (z-score) for competition was 58.999. More than 40 years ear- to his peers because his z-score z =( 1.83)
the Washington Nationals, who won 93 games (and lier, Romanian gymnast Nadia Comaneci took the
(
the World Series!). Interpret this value. world by storm with the first perfect 10. With an was greater than Rebecca’s z =1.43).
12. Stand tall The heights of the 25 students in Mrs. overall score of 79.275, Comaneci also won the 58.999 −54.719
5
all-around gold medal. Because the scoring sys-
Nataro’s statistics class have a mean of 67 inches 17. Biles: z = = 2.38
and a standard deviation of 4.29 inches. Find the tems have changed, these two scores aren’t directly 1.8
standardized score (z-score) for Boris, a member of comparable. In 2019, the 23 gymnasts who com-
pleted the all-around had a mean score of 54.719
the class who is 75 inches tall. Interpret this value. 79.275 −76.527
points and a standard deviation of 1.800 points. In Comaneci: z = = 2.07
13. Where are the old folks? Based on data from the 2016 1976, the top 24 gymnasts in the all-around had a 1.327
Current Population Survey, the percent of residents mean score of 76.527 points and a standard devi-
aged 65 or older in the 50 states and the District of ation of 1.327 points. Which gymnast had a bet- Biles had a better performance, relatively
Columbia has mean 15.62% and standard deviation ter performance, relatively speaking? Explain your speaking, because her z-score z =( 2.38)
1.83%. 3 reasoning. was greater than Comaneci’s z-score
(a) Find and interpret the standardized score (z-score) 18. Comparing batting averages Three landmarks of
for the state of Colorado, which had 13.4% of its baseball achievement are Ty Cobb’s batting average ( z = 2.07).
residents aged 65 or older. of 0.420 in 1911, Ted Williams’s 0.406 in 1941, and
(b) The standardized score for Florida is =z 2.34. Find the George Brett’s 0.390 in 1980. These batting averages 18. Cobb: z = 0.420 −0.266 = 4.15
percent of the state’s residents that were 65 or older. cannot be compared directly because the distribution 0.0371
0.406 −0.267
Williams: z = = 4.26
0.0326
=
8. (a) 9/250.36 36thpercentile
=
03_StarnesSPA4e_24432_ch02_088_153.indd 95 93 −81 07/09/20 1:54 PM 0.390 −0.261
(b) Asher’s height is greater than 88% 11. z = 15.9 =.075. The Washington Brett: z = 0.0317 = 4.07
of the 25 students (0.882522students) Nationals number of wins in 2019 is Williams had the best performance,
=
×
in Mrs. Nataro’s statistics class. Asher is 0.75 standard deviations above the mean of relatively speaking, because his z-score
74 inches tall. 81 wins. ( z = 4.26) was greater than both Cobb’s
9. (a) 17/20 0.85 85thpercentile 75 −67 ( z = 4.15) and Brett’s z = 4.07)z-scores.
=
=
(
×
=
(b) 0.25 205, so Luis has more pairs of 12. z = 4.29 =1.86. Boris has a height
shoes than 5 of the 20 male students. Luis has that is 1.86 standard deviations above the
7 pairs of shoes. mean of 67 inches.
=
10. (a) 6/200.3 30thpercentile 13.4 −15.62
=
(b) 0.75 × 20 15, so the selling price 13. (a) z = 1.83 =−1.21.
=
of the phone is greater than 15 of the 20 Colorado has a percent of residents aged 65
phones. This phone has a selling price of $425. or older that is 1.21 standard deviations below
the mean of 15.62%.
LESSON 2.1 • Describing Location in a Distribution 95
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