SPA 3e_ Teachers Edition _ Ch 6

L E S S O N 6.1 • What Is a Sampling Distribution? 407
than 64 inches, the class simulated 200 SRSs of size
n 5 20 from a normal population with mean
m 5 64 inches and standard deviation s 5 2.5
inches. Here are the results of the simulation.
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62.5 63.0 63.5 64.0 64.5 65.0 65.5 66.0
Sample mean height (in.)
(a) There is one dot on the graph at 62.5. Explain what
this dot represents.
(b) Would it be unusual to get a sample mean of 64.7
or more in a sample of size 20 when m 5 64?
Explain.
(c) Based on your answer to part (b), is there convincing
evidence that the mean height of the population
of 16-year-old girls at this school is greater than 64
inches? Explain.
12. Relying on bathroom scales A manufacturer of
bathroom scales says that when a 150-pound
weight is placed on a scale produced in the factory,
the weight indicated by the scale is normally
distributed with a mean of 150 pounds and a
standard deviation of 2 pounds. A consumeradvocacy
group acquires an SRS of 12 scales from
the manufacturer and places a 150-pound weight
on each one. The group gets a mean weight of
149.1 pounds, which makes them suspect that
the scales underestimate the true weight. To test
this, they use a computer to simulate 200 samples
of 12 scales from a population with a mean of
150 pounds and standard deviation of 2 pounds.
Here is a dotplot of the means from these 200
samples.
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148.0 148.5 149.0 149.5 150.0 150.5 151.0 151.5
Sample mean weight (lb)
(a) There is one dot on the graph at 151.2. Explain
what this dot represents.
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(b) Would it be unusual to get a sample mean of 149.1 or
less in a sample of size 12 when m 5 150? Explain.
(c) Based on your answer to part (b), is there convincing
evidence that the scales produced by this manufacturer
underestimate true weight? Explain.
Applying the Concepts
13. Instant winners A fast-food restaurant promotes
certain food items by giving a game piece with
each item. Advertisements proclaim that “25% of
the game pieces are Instant Winners!” To test this
claim, a frequent diner collects 20 game pieces and
gets only 3 instant winners.
(a) Identify the population, the parameter, the sample,
and the statistic in this context.
Suppose the advertisements are correct and
p 5 0.25. The dotplot shows the distribution of the
sample proportion of instant winners in 100 simulated
SRSs of size n 5 20.
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0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6
Sample proportion of instant winners
(b) Would it be unusual to get a sample proportion of
p^ 5 3/20 5 0.15 or less in a sample of size n 5 20
when p 5 0.25? Explain.
(c) Based on your answer to part (b), is there convincing
evidence that fewer than 25% of all game
pieces are instant winners? Explain.
14. Puny guppies? A large pet store that specializes
in tropical fish has several thousand guppies. The
store claims that the lengths of its guppies are
approximately normally distributed with a mean of
5 centimeters and a standard deviation of 0.5 centimeter.
You come to the store and buy 10 randomly
selected guppies and find that the mean length of
your 10 guppies is only 4.8 centimeters.
(a) Identify the population, the parameter, the sample,
and the statistic in this context.
Suppose the store’s description of the lengths of
its guppies is true. The dotplot on the next page
shows the distribution of sample means from 200
simulated SRSs of size n 5 10 from a normally distributed
population with m 5 5 centimeters and
s 5 0.5 centimeter.
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11. (a) In one SRS of size n 5 20, the
mean height was 62.5 inches.
(b) No, in the 200 simulated samples,
23 of the SRSs had a mean of 64.7
or more. Based on the simulation,
P( x ≥ 64.7) = 23∙200 = 0.115.
(c) No; because the probability from
part (b) isn’t small, it is plausible that the
true mean is m 5 64 and the class got
a sample mean of x 5 64.7 by chance
alone.
12. (a) In one SRS of size n 5 12, the
mean weight was 151.2 pounds.
(b) No; in the 200 simulated samples,
22 of the SRSs had a mean of
149.1 or less. Based on the simulation,
P( x ≤ 149.1) = 22∙200 = 0.11.
(c) No; because the probability from
part (b) isn’t small, it is plausible that the
true mean is m 5 150 and the group got
a sample mean of x 5 149.1 by chance
alone.
13. (a) Population: All game pieces.
Parameter: p 5 the true proportion of
the population that are instant winners.
Sample: The 20 game pieces collected by
the frequent diner. Statistic: The proportion
of the sample that are instant winners;
p^ 5 3/20 5 0.15.
(b) No; in the 100 simulated samples,
18 of the SRSs had a sample proportion
of 0.15 or lower. Based on the simulation,
P( p^ ≤ 0.15) = 18∙100 = 0.18.
(c) No; because the probability from
part (b) isn’t small, it is plausible that
the true proportion is p 5 0.25 and the
frequent diner got a sample proportion
of p^ 5 0.15 by chance alone.
Lesson 6.1
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Answers continued
10. (a) In one SRS of size n 5 60, 36.7% of
the first serves were successful.
(b) Yes; in the 200 simulated samples, only
1 of the SRSs had a sample proportion of
0.70 or higher. Based on the simulation,
P( p^ ≥ 0.70) = 1∙200 = 0.005.
(c) Yes; because the probability from part
(b) is small, it is not plausible that the true
proportion is still p 5 0.53 and the player got
a sample proportion of p^ 5 0.70 by chance
alone.
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