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LESSON 2.4 • The Empirical Rule and Assessing Normality 125

Exercises 12. The 84th percentile of this
distribution is 71.5 inches. Approximately
Building Concepts and Skills (a) The proportion of 14-year-old boys with a choles- 84% of adult American men are shorter
terol level less than 125 mg/dl
1. Give one reason that normal distributions are than 71.5 inches
important in statistics. (b) The percent of 14-year-old boys with cholesterol
levels between 200 and 225 mg/dl 13. About 2.5%
2. The empirical rule is also called the Lesson 2.4
rule. 11. When the chips are down Refer to Exercise 7. What 14. About 0.15%
pg 121 bag weight is at the 84th percentile of the distribu-
3. True/False: The empirical rule applies to any distri- tion? Justify your answer.
bution of quantitative data. 15. The dotplot shows that the
12. Tall men Refer to Exercise 8. What height is at the
4. Give an example of a quantitative variable with a 84th percentile of the distribution? Justify your distribution of EPA highway gas mileage
distribution that is approximately normal. answer. estimates is skewed to the right.
5. Give an example of a quantitative variable that has 13. Ace! Refer to Exercise 9. About 2.5% of Rafa’s first
a non-normal distribution. serves are faster than what speed? Justify your answer. 16. The histogram shows that the
(C) 2021 BFW Publishers -- for review purposes only.
6. True/False: If a graph of quantitative data looks 14. Low cholesterol Refer to Exercise 10. About 0.15% distribution of the percent of foreign-
roughly symmetric, single-peaked, and bell- of 14-year-old boys have cholesterol levels less than born residents in each of the 50 states is
shaped, then the distribution is approximately what value? Justify your answer.
normal. strongly skewed to the right.
15. Normal highway driving? The dotplot shows the
Mastering Concepts and Skills EPA highway gas mileage estimates in miles per 17. The histogram looks roughly
gallon (mpg) for a sample of 21 model year 2020 symmetric, single-peaked, and
7. How much is 9 ounces? The weights of 9-ounce midsize cars. Explain why this distribution of
14
pg 120 bags of a particular brand of potato chips can highway gas mileage is not approximately normal. somewhat bell-shaped.
be modeled by a normal distribution with mean
±
µ = 9.12 ounces and standard deviation σ = 0.05 dd Mean ±1SD: 54.98 1(6.57) 48.41to 61.55
dd
ounce. Use the empirical rule to answer the fol- dd
lowing questions. d dd dd d dd d d d d 32out of 45 = 71.1%
dd d
±
(a) What percent of bags weigh less than 9.07 ounces? 25 30 35 40 45 50 Mean ± 2SD: 54.98 2(6.57) 41.84 to 68.12
(b) What proportion of bags weigh between 8.97 and Highway gas mileage (mpg) 43out of 45 = 95.6%
9.17 ounces?
±
16. Normal to be foreign born? The histogram displays Mean ± 3SD: 54.98 3(6.57) 35.27 to 74.69
8. Modeling men’s heights The heights of adult Amer- the percent of foreign-born residents in each of the
15
ican men can be modeled by a normal distribution 50 states in 2018. Explain why this distribution is 45out of 45 =100%
with mean µ = 69 inches and standard deviation not approximately normal. These percentages are fairly close to
σ = 2.5 inches. Use the empirical rule to answer the
following questions. 15 the 68%, 95%, and 99.7% targets for
(a) What percent of adult American men are greater a normal distribution. The graphical
than 74 inches tall? 10
(b) What proportion of adult American men have Number of states and numerical evidence suggests that
heights between 64.0 and 71.5 inches? 5 the distribution of ages at presidential
9. Rafa serves again! Tennis superstar Rafael Nadal’s inauguration is approximately normal.
first-serve speeds in a recent season can be modeled 0
by a normal distribution with mean 115 mph and 0 5 10 15 20 25 30
standard deviation 6 mph. Use the empirical rule to Percent foreign-born residents
approximate the following:
(a) The proportion of Rafa’s first serves that were 17. Too young to be president? Here are the ages of the
faster than 121 mph pg 123 first 45 U.S. presidents when they took office.
(b) The percent of Rafa’s first serves with speeds
between 109 and 133 mph 42 43 46 46 47 47 48 49 49 50 51 51 51 51 51
52 52 54 54 54 54 54 55 55 55 55 56 56 56 57
10. Cholesterol modeled Cholesterol levels for teen-
age boys can be modeled by a normal distribu- 57 57 57 58 60 61 61 61 61 64 64 65 68 69 70
tion with mean 150 mg/dl and standard deviation
25mg/dl. Use the empirical rule to approximate A histogram and summary statistics for the age
the following: data are shown in the following display. Is the
03_StarnesSPA4e_24432_ch02_088_153.indd 125 5. Number of siblings 07/09/20 1:56 PM
FULL SOLUTIONS TO LESSON 2.4
EXERCISES 6. False. Even if a graph of the data looks
roughly symmetric and bell-shaped, we
You can find the full solutions for this lesson shouldn’t assume that the distribution is
by clicking on the link in the TE-book or by approximately normal.
logging into the teachers’ resources on our 7. (a) About 16%
digital platform.
=
+
(b) About15.85% 68% 83.85% 0.8385
=
Answers to Lesson 2.4 Exercises 8. (a) About 2.5%
(b) About13.5% 68% 81.5% 0.815
=
+
=
1. Normal distributions are good descriptions
=
for some distributions of real data. 9. (a) About16% 0.16
=
2. 68–95–99.7 (b) About 68% +15.85% 83.85%
=
3. False. The empirical rule only applies to 10. (a) About16% 0.16
(b) About2.35% 0.0235
=
distributions of quantitative data that are
approximately normal. 11. The 84th percentile of this distribution is
4. SAT scores 9.17 ounces. Approximately 84% of 9-ounce
bags of chips weigh less than 9.17 ounces.
LESSON 2.4 • The Empirical Rule and Assessing Normality 125
03_TysonTEspa4e_25177_ch02_088_153_4pp.indd 125 10/11/20 7:46 PM