SPA 3e_ Teachers Edition _ Ch 6

L E S S O N 6.3 • The Sampling Distribution of a Sample Count
423
18/08/16 5:01 PMStarnes_3e_CH06_398-449_Final.indd 423
appear in the 100,000 boxes of cereal that are part
of this promotion. You buy 12 boxes and let X 5
the number of Kyle Busch cards in the sample.
(a) Calculate the mean and standard deviation of the
sampling distribution of X.
(b) Interpret the standard deviation from part (a).
4. What, me marry? In the United States, 20% of
adults ages 25 and older have never been married,
more than double the figure recorded for 1960. 7
Select a random sample of 50 U.S. adults ages 25
and older and let Y 5 the number of individuals in
the sample who have never married.
(a) Calculate the mean and standard deviation of the
sampling distribution of Y.
(b) Interpret the standard deviation from part (a).
5. Are lefties normal? Refer to Exercise 1. Would it be
pg 420 appropriate to use a normal distribution to model
the sampling distribution of X 5 the number of
left-handed students in the sample? Justify your
answer.
6. Is hip dysplasia normal? Refer to Exercise 2. Would it
be appropriate to use a normal distribution to model
the sampling distribution of Y 5 the number of Labs
with dysplasia in the sample? Justify your answer.
7. Is NASCAR normal? Refer to Exercise 3. Would
it be appropriate to use a normal distribution to
model the sampling distribution of X 5 the number
of Kyle Busch cards in the sample? Justify your
answer.
8. A normal marriage? Refer to Exercise 4. Would
it be appropriate to use a normal distribution to
model the sampling distribution of Y 5 the number
of individuals in the sample who have never married?
Justify your answer.
9. Lefties are all right Refer to Exercises 1 and 5.
pg 421 Calculate the probability that at least 15 of the
members of the sample are left-handed.
10. Never been married Refer to Exercises 4 and 8.
Calculate the probability that at most 5 of the individuals
in the sample have never been married.
11. Public transportation In a large city, 34% of residents
use public transportation at least once per
week. If the mayor selects a random sample of 200
residents, calculate the probability that at most 60
residents in the sample use public transportation at
least once per week.
12. U.S. quarters According to www.usmint.gov, 54%
of the quarters minted in 2014 were produced by
the U.S. Mint in Denver, Colorado (the rest were
produced in Philadelphia). In a random sample of
200 quarters, what is the probability that at least
115 of them were minted in Denver?
8. Yes; because np = 50(0.20) = 10 $ 10
and n(1− p) = 50(1− 0.20)= 40 ≥ 10, the
sampling distribution of Y is approximately
normal.
9. From Exercises 1 and 5, X is approximately
normal with a mean of 11 and standard
deviation of 3.13.
15 − 11
z = ≈ 1.28; P(X ≥ 15)≈ P(Z ≥ 1.28)
3.13
= 1− 0.8997 = 0.1003
Using technology: Applet/normalcdf(lower:15,
upper:1000, mean:11, SD:3.13) 5 0.1006
10. From Exercises 4 and 8, Y is
approximately normal with a mean of 10 and
standard deviation of 2.83.
z = 5 − 10
2.83 ≈ −1.77;
P(Y ≤ 5) ≈ P(Z ≤ −1.77) = 0.0384
Applying the Concepts
13. Tasty chips For a statistics project, Zenon decided
to investigate if students at his school prefer namebrand
potato chips to store-brand potato chips. He
prepared two identical bowls of chips, filling one
with name-brand chips and the other with storebrand
chips. Then, he selected a random sample
of 30 students, had each student try both types of
chips in random order, and recorded which type of
chip each student preferred. Assume that 50% of
students at Zenon’s school prefer the name-brand
chips. Let X 5 the number of students in the sample
that prefer the name-brand chips. 8
(a) Calculate the mean and standard deviation of the
sampling distribution of X. Interpret the standard
deviation.
(b) Justify that the distribution of X is approximately
normal.
(c) Calculate the probability that 19 or more of the
students will prefer the name-brand chips.
14. Blood types About 10% of people in the United
States have type B blood. Suppose we take a random
sample of 120 U.S. residents, and let X 5 the number
of residents in the sample who have type B blood.
(a) Calculate the mean and standard deviation of the
sampling distribution of X. Interpret the standard
deviation.
(b) Justify that the distribution of X is approximately
normal.
(c) Calculate the probability that 16 or more individuals
in the sample have type B blood.
15. More chips! Refer to Exercise 13. In Zenon’s study,
19 of the 30 students chose the name-brand chips.
Based on your answer to Exercise 13(c), does this
provide convincing evidence that more than half of
the students at Zenon’s school prefer name-brand
potato chips? Explain.
16. More on blood type Refer to Exercise 14. Some people
believe that one’s blood type has an impact on personality.
For example, people with type B blood are
supposed to be more creative, active, and passionate.
To test this hypothesis, Jason selects a random sample
of 120 art, music, and drama majors at his college
and finds that 16 of them have type B blood. Based on
your answer to Exercise 14(c), does this provide convincing
evidence that art, music, and drama majors at
Jason’s college are more likely than the general population
to have type B blood? Explain.
Extending the Concepts
17. Binomial transportation Refer to Exercise 11. Use
a binomial distribution to calculate the probability
that at most 60 residents in the sample use public
transportation at least once per week. Hint: See
Lesson 5.4.
18/08/16 5:01 PM
Using technology: Applet/normalcdf(lower:
−1000, upper:5, mean:10, SD:2.83) 5 0.0386
11. X 5 the number of residents who use
public transportation at least once per week.
Mean: m X = np = 200(0.34) = 68;
SD: s X = "np(1 − p)
= "200(0.34)(1 − 0.34) = 6.70
Shape: Approximately normal because
np = 200(0.34) = 68 ≥ 10 and
n(1−p) = 200(1− 0.34) = 132 ≥ 10.
60 − 68
z = ≈ −1.19;
6.70
P(X ≤ 60) ≈ P(Z ≤ −1.19) = 0.1170
Using technology: Applet/normalcdf
(lower:−1000, upper:60, mean:68,
SD:6.70) 5 0.1162
12. X 5 the number of quarters minted
in Denver.
Mean: m X = np = 200(0.54) = 108;
SD: s X = "np(1 − p)
= "200(0.54)(1 − 0.54) = 7.05
Shape: Approximately normal because
np = 200(0.54) = 108 ≥ 10 and
n(1− p) = 200(1− 0.54) = 92 ≥ 10.
115 − 108
z = ≈ 0.99; P(X ≥ 115)
7.05
≈ P(Z ≥ 0.99) = 1− 0.8389 = 0.1611
Using technology: Applet/normalcdf
(lower:115, upper:1000, mean:108,
SD:7.05) 5 0.1604
13. (a) m X = np = 30(0.5) = 15
students; s X = "np(1− p)
= "30(0.5)(1 − 0.5) = 2.74 students.
If many samples of size 30 were taken, the
number of students who prefer namebrand
chips would typically vary by about
2.74 from the mean of 15.
(b) Because np = 30(0.5) = 15 ≥ 10
and n(1 − p) = 30(1 − 0.5) = 15 ≥ 10,
the sampling distribution of X is
approximately normal.
19 − 15
(c) z = ≈ 1.46; P(X ≥ 19) ≈
2.74
P(Z ≥ 1.46) = 1− 0.9279 = 0.0721
Using technology: Applet/normalcdf
(lower:19, upper:1000, mean:15,
SD:2.74) 5 0.0722
14. (a) m X = np = 120(0.1) = 12
residents; s X = "np(1− p)
= "120(0.1)(1 − 0.1) = 3.29 residents.
If many samples of size 120 were
taken, the number of residents who have
type B blood would typically vary by
about 3.29 from the mean of 12.
(b) Because np = 120(0.1) = 12 $ 10
and n(1− p) = 120(1 − 0.1) = 108 $ 10,
the sampling distribution of X is
approximately normal.
16 − 12
(c) z = ≈ 1.22; P(X ≥ 16) ≈
3.29
P(Z ≥ 1.22) = 1 − 0.8888 = 0.1112
Using technology: Applet/normalcdf
(lower:16, upper:1000, mean:12,
SD:3.29) 5 0.112
15. No; assuming that 50% of students
prefer name-brand chips, there’s about a
7% chance that the number of students
who prefer name-brand chips (out of
an SRS of 30) is 19 or more. The results
from Zenon’s study could have happened
purely by chance, so we do not have
convincing evidence that more than half
of the students prefer name-brand chips.
Answers 16–17 are on page 424
L E S S O N 6.3 • The Sampling Distribution of a Sample Count 423
Lesson 6.3
Starnes_3e_ATE_CH06_398-449_v3.indd 423
11/01/17 3:55 PM