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LESSON 2.5 • Normal Distributions: Finding Areas from Values 137

Lesson 2.5 13. (i) z = (9.2 9.12)/0.051.60; the
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WHA T DID Y OU LEARN ? proportion of z-scores greater than
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LEARNING TARGET EXAMPLES EXERCISES z =1.60 is1–0.9452 0.0548.
Find the proportion of values to the left of a boundary in a normal distribution. p. 131 9–12 (ii) Applet/normalcdf(lower:9.2,
Find the proportion of values to the right of a boundary in a normal distribution. p. 133 13–16 upper:1000,mean: 9.12,SD: 0.05) = Lesson 2.5
0.0548. The proportion of 9-ounce bags of
Find the proportion of values between two boundaries in a normal distribution. p. 135 17–20
this brand of potato chips that weigh more
than 9.2 ounces is approximately 0.0548.
Exercises 14. (i) z = (0.3 0.261)/0.034 1.15;
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the proportion of z-scores greater than
Building Concepts and Skills of potato chips weigh less than the advertised z =1.15is1–0.8749 0.1251.
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(C) 2021 BFW Publishers -- for review purposes only.
9 ounces? Is this likely to pose a problem for the
1. The standard normal distribution has a mean company that produces these chips?
of and a standard deviation of (ii) Applet/normalcdf(lower:0.3,
. 10. On the bench In baseball, a player’s batting average
is the proportion of times the player gets a hit out of upper:1000,mean: 0.261,SD: 0.034) =
2. True/False: The entry for each z-score in Table A is the the total number of times at bat. The distribution of 0.1257. The proportion of Major
area under the standard normal curve to the left of z. batting averages in a recent season for Major League
3. What is the first step when finding areas in any Baseball players with at least 100 plate appearances League Baseball players that have a
normal distribution? can be modeled by a normal distribution with mean batting average of 0.300 or higher is
µ = 0.261 and standard deviation σ = 0.034. A
4. When calculating an area in a normal distribution, player with a batting average below 0.200 is at risk of approximately 0.1257.
one option is to and then find the sitting on the bench during important games. About
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area using Table A or technology. what percent of players are at risk? 15. (i) z = (730 538)/117 1.73; the
5. In the standard normal curve, the area to the left of 11. Lower SATs In the class of 2019, more than 1.6 proportion of z-scores greater than
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z = 0.53 is 0.7019. What is the area to the right of million students took the SAT. The distribution of z =1.73is1–0.9582 0.0418.
z = 0.53? Explain your answer. scores on the math section (out of 800) is approx-
6. True/False: When finding area in a normal distribu- imately normal with a mean of 528 and standard (ii) Applet/normalcdf(lower: 730,
tion, it matters whether the boundary value is included deviation of 117. About what proportion of stu- upper:1000,mean: 528,SD:117)0.0421.
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x
or not—for example, <x 120 versus ≤ 120. dents who took the SAT scored less than 350 on the
math section? About 4.2% of students who took the
7. Find the proportion of observations in a standard
normal distribution that satisfies each of the fol- 12. Blood pressure According to a health information SAT math test meet the University of
lowing statements. website, the distribution of adults’ diastolic blood Michigan requirement (at least 730).
(a) z < 2.46 pressure (in millimeters of mercury) can be modeled
by a normal distribution with mean 70 and stan-
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(b) z >− 1.66 dard deviation 20. A healthy diastolic pressure for 16. (i) z = (120 70)/202.50; the
(c) z is between 0.50 and 1.79 an adult is less than 80. About what proportion of proportion of z-scores greater than
adults have healthy diastolic blood pressures?
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8. Find the proportion of observations in a standard z = 2.50 is1–0.9938 0.0062.
normal distribution that satisfies each of the fol- 13. Chips galore Refer to Exercise 9. What proportion
lowing statements. pg 133 of 9-ounce bags of this brand of potato chips weigh (ii) Applet/normalcdf(lower:120,
(a) z <− 1.39 more than 9.2 ounces? upper:1000,mean: 70,SD: 20)0.0062.
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(b) z > 2.15 14. Batter up! Refer to Exercise 10. What proportion of About 0.62% of adults have a diastolic
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(c) z is between 1.11 and 0.32 players have a batting average of 0.300 or higher?
15. SATisfactory for admission Refer to Exercise 11. blood pressure that is classified as
Mastering Concepts and Skills The University of Michigan has a recommended SAT hypertensive crisis (greater than 120 mm
math score of at least 730. What percent of students
9. Weighing potato chips The weights of 9-ounce bags who took the SAT math test meet this requirement? of mercury).
pg 131 of a particular brand of potato chips can be mod-
eled by a normal distribution with mean µ = 9.12 16. Borderline blood pressure Refer to Exercise 12.
ounces and standard deviation σ = 0.05 ounce. Diastolic blood pressures higher than 120 are clas-
About what percent of 9-ounce bags of this brand sified as a hypertensive crisis and require emergency
03_StarnesSPA4e_24432_ch02_088_153.indd 137 07/09/20 1:57 PM
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9. (i) z = (9 9.12)/0.05 =−2.40: the have batting averages less than 0.200 and
proportion of z-scores less than z = –2.40 is are at risk of sitting on the bench during
0.0082. important games.
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(ii) Applet/normalcdf(lower:–1000, 11. (i) z = (350 528)/117 =−1.52; the
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upper: 9, mean:9.12, SD:0.05) 0.0082. proportion of z-scores less than z = –1.52 is
About 0.82% of 9-ounce bags of potato chips 0.0643.
weigh less than 9 ounces. Because only a (ii) Applet/normalcdf(lower:–1000,
small proportion of 9-ounce bags of potato upper: 350, mean: 528,SD:117)0.0641.
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chips weigh less than the advertised 9 ounces, The proportion of students who took the SAT
it is unlikely that this will be a problem for the and scored less than 350 on the math section
company that produces these chips. is about 0.0641.
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10. (i) z = (0.2 0.261)/0.034 =−1.79; the 12. (i) z = (80 70)/200.50; the proportion
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proportion of z-scores less than z =−1.79 is of z-scores less than z = 0.50 is 0.6915.
0.0367. (ii) Tech: Applet/normalcdf(lower:–1000,
(ii) Tech: Applet/normalcdf(lower:–1000, upper: 80,mean: 70,SD: 20)0.6915. The
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upper: 0.2, mean:0.261,SD: 0.034) =0.0364. proportion of adults who have healthy
About 3.6% of Major League Baseball players diastolic blood pressures is about 0.692.
LESSON 2.5 • Normal Distributions: Finding Areas from Values 137
03_TysonTEspa4e_25177_ch02_088_153_4pp.indd 137 10/11/20 7:47 PM