SPA 3e_ Teachers Edition _ Ch 6

446
C H A P T E R 6 • Sampling Distributions
Answers to Chapter 6 Review
Exercises
1. Population: All eggs shipped in one
day. Sample: The 200 eggs examined.
Parameter: The proportion p of eggs
shipped that day that had salmonella.
Statistic: The proportion of eggs in
the sample that had salmonella,
p^ = 9
200 = 0.045.
2. (a)
Sample #1: 64, 66, 71 Median 5 66
Sample #2: 64, 66, 73 Median 5 66
Sample #3: 64, 66, 76 Median 5 66
Sample #4: 64, 71, 73 Median 5 71
Sample #5: 64, 71, 76 Median 5 71
Sample #6: 64, 73, 76 Median 5 73
Sample #7: 66, 71, 73 Median 5 71
Sample #8: 66, 71, 76 Median 5 71
Sample #9: 66, 73, 76 Median 5 73
Sample #10: 71, 73, 76 Median 5 73
d
d
d
d
d
d
66 67 68 69 70 71 72 73
Sample median book length
66 + 66 + 66 + 71 + 71 +
71 + 71 + 73 + 73 + 73
(b) m median =
10
= 701
10 = 70.1
The sample median is a biased estimator
of the population median. The mean
of the sampling distribution is equal to
70.1, which is less than the value of the
population median of 71.
(c) The sampling distribution of the
sample median will be less variable
because the sample size is larger. The
estimated median book length will
typically be closer to the true median
book length. In other words, the estimate
will be more precise.
3. (a) m X = np = 500(0.24) = 120
people; s X = "np(1 − p)
= "500(0.24)(1 − 0.24) = 9.55 people
(b) If many samples of size 500 were
taken, the number of people who are
under 18 years old would typically vary
by about 9.55 from the mean of 120.
(c) The sampling distribution of X
is approximately normal because
np = 500(0.24) = 120 ≥ 10 and
n(1 − p) = 500(1 − 0.24) = 380 ≥ 10
j
j
j
We can use sampling distributions to determine
what values of a statistic are likely to happen by
chance alone and how much a statistic typically
varies from the parameter it is trying to estimate.
A statistic used to estimate a parameter is an
unbiased estimator if the mean of its sampling
distribution is equal to the value of the parameter
being estimated. That is, the statistic doesn’t
consistently overestimate or consistently underestimate
the value of the parameter when many
random samples are selected.
The sampling distribution of any statistic will
have less variability when the sample size is larger.
That is, the statistic will be a more precise estimator
of the parameter with larger sample sizes.
Sample Counts and Sample Proportions
j
j
Let X 5 the number of successes in a random sample
of size n from a large population with proportion
of successes p. The sampling distribution of a
sample count X describes the distribution of values
taken by the sample count X in all possible samples
of the same size from the same population.
j The mean of the sampling distribution of X
is m X = np. The mean describes the average
value of X in repeated random samples.
j The standard deviation of the sampling distribution
of X is s X = !np(1 − p). The standard
deviation describes how far the values of X typically
vary from m X in repeated random samples.
j The shape of the sampling distribution of X
will be approximately normal when the Large
Counts condition is met: np ≥ 10 and n(1 2 p)
≥10.
Let p^ 5 the proportion of successes in a random
sample of size n from a large population with proportion
of successes p. The sampling distribution of a
sample proportion p^ describes the distribution of values
taken by the sample proportion p^ in all possible
samples of the same size from the same population.
j The mean of the sampling distribution of p^ is
m p^ = p. The mean describes the average value
of p^ in repeated random samples.
j The standard deviation of the sampling
p(1 − p)
distribution of p^ is s p^ 5 . The
Å n
Starnes_3e_CH06_398-449_Final.indd 446
100 − 120
(d) z = = −2.09;
9.55
110 − 120
z = = −1.05
9.55
P(100 ≤ X ≤ 110) ≈ P(−2.09 ≤
Z ≤ − 1.05) = 0.1469 − 0.0183 = 0.1286
Using technology: Applet/normalcdf
(lower:100, upper:110, mean:120, SD:9.55)
5 0.1294
j
standard deviation describes how far the values
of p^ typically vary from p in repeated random
samples.
The shape of the sampling distribution of
p^ will be approximately normal when the
Large Counts condition is met: np ≥ 10 and
n(1 2 p) ≥ 10.
Sample Means
j
j
Let x 5 the mean of a random sample of size n
from a large population with mean m and standard
deviation s. The sampling distribution of
a sample mean x describes the distribution of
values taken by the sample mean x in all possible
samples of the same size from the same
population.
j The mean of the sampling distribution of x is
m x = m. The mean describes the average value
of x in repeated random samples.
j The standard deviation of the sampling distribution
of x is s− x = s . The standard deviation
describes how far the values of x typically
"n
vary from m in repeated random samples.
The shape of the sampling distribution of
x will be approximately normal when the
Normal/Large Sample condition is met: The
population is normal or the sample size is large
(n ≥ 30). The fact that the sampling distribution
of x becomes approximately normal—
even when the population is non-normal—as
the sample size increases is called the central
limit theorem.
Probability Calculations
j
j
When the sampling distribution of a statistic is
approximately normal, you can use z-scores and
Table A or technology to do probability calculations
involving the statistic.
To determine which sampling distribution to use,
consider whether the variable of interest is categorical
or quantitative. If it is categorical, use the
sampling distribution of a sample count X or the
sampling distribution of a sample proportion p^ . If
it is quantitative, use the sampling distribution of
a sample mean x.
TRM chapter 6 Review Exercise Videos
Video solutions to the Chapter 6 Review
Exercises are available to teachers and
students. Access them by clicking on the link
in the TE-book, logging into the Teacher’s
Resource site, or accessing this resource
on the TRFD.
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C H A P T E R 6 • Sampling Distributions
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