SPA 3e_ Teachers Edition _ Ch 6

Chapter 6 Practice Test
449
8. Scores on the mathematics part of the SAT exam in a
recent year followed a normal distribution with mean
515 and standard deviation 114. You choose an SRS
of 100 students and calculate x5 mean SAT Math
score. Which of the following are the mean and standard
deviation of the sampling distribution of x?
(a) Mean 5 515, SD 5 114
114
(b) Mean 5 515, SD 5
"100
(c) Mean 5 515 114
, SD 5
100 100
(d) Mean 5 515
100 , SD 5 114
"100
9. In a congressional district, 55% of the registered voters
are Democrats. Which of the following is closest to
the probability of getting less than 50% Democrats in
a random sample of size 100?
(a) 0.157 (b) 0.496 (c) 0.504 (d) 0.843
10. A statistic is an unbiased estimator of a parameter
when
(a) the statistic is calculated from a random sample.
(b) in all possible samples of a specific size, the distribution
of the statistic has a shape that is approximately
normal.
(c) in all possible samples of a specific size, the values of
the statistic are very close to the value of the parameter.
(d) in all possible samples of a specific size, the values of
the statistic are centered at the value of the parameter.
Section II: Free Response
11. Here are histograms of the values taken by three
sample statistics in several hundred samples from the
same population. The true value of the population
parameter is marked with an arrow on each histogram.
Which statistic would provide the best estimate
of the parameter? Explain.
A B C
12. The amount that households pay service providers for
access to the Internet varies quite a bit, but the mean
monthly fee is $48 and the standard deviation is $20.
The distribution is not normal: Many households pay
a base rate for low-speed access, but some pay much
more for faster connections. A sample survey asks an
SRS of 500 households with Internet access how much
they pay per month. Let x be the mean amount paid
by the members of the sample.
(a) Calculate the mean and standard deviation of the sampling
distribution of x. Interpret the standard deviation.
(b) What is the shape of the sampling distribution of x?
Justify.
(c) Find the probability that the average amount paid by
the sample of households exceeds $50.
13. According to government data, 22% of American children
under the age of 6 live in households with incomes
less than the official poverty level. A study of learning in
early childhood chooses an SRS of 300 children.
(a) Let X 5 the count of children in this sample who live in
households with incomes less than the official poverty
level. What is the shape of the sampling distribution of
X? Justify your answer.
(b) Find the probability that more than 20% of the sample
are from poverty-level households.
8. b
9. a
10. d
11. Statistic A. Both statistics A and B
appear to be unbiased, with the center of
their sampling distributions equal to the
value of the parameter, but statistic A has
less variability than statistic B.
12. (a) m x = m = $48;
s x = s !n = 20
!500 = $0.89
In SRSs of size n 5 500, the sample mean
amount paid for Internet will typically
vary by about $0.89 from the true mean
of $48.
(b) Because n = 500 ≥ 30, the sampling
distribution of x is approximately normal
by the central limit theorem.
50 − 48
(c) z = = 2.25; P( x > 50)
0.89
= P(Z > 2.25) = 1 − 0.9878 = 0.0122
Using technology: Applet/normalcdf
(lower:50, upper:1000, mean:48, SD:0.89)
5 0.0123
13. (a) The sampling distribution of
X is approximately normal because
np = 300(0.22) = 66 ≥ 10 and
n(1 − p) = 300(1 − 0.22) = 234 ≥ 10.
(b) 20% of 300 is 60, so we want to find
P( X > 60).
m X = np = 300(0.22) = 66;
s X = "np(1 − p)
= "300(0.22)(1 − 0.22) = 7.17
60 − 66
z = = −0.84;
7.17
P(X > 60) = P(Z > −0.84) = 0.7995
Using technology: Applet/normalcdf
(lower:60, upper:1000, mean:66, SD:7.17)
5 0.7987
Practice Test
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C H A P T E R 6 • Practice Test 449
Starnes_3e_ATE_CH06_398-449_v3.indd 449
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