324 CHAPTER 9 TESTS OF HYPOTHESES FOR A SINGLE SAMPLE9-73. An inspector of flow metering devices used to admin-
ister fluid intravenously will perform a hypothesis test to
determine whether the mean flow rate is different from the flow
rate setting of 200 milliliters per hour. Based on prior
information the standard deviation of the flow rate is assumed
to be known and equal to 12 milliliters per hour. For each of the
following sample sizes, and a fixed 0.05, find the probabil-
ity of a type II error if the true mean is 205 milliliters per hour.
(a)n 20
(b)n 50
(c)n 100
(d) Does the probability of a type II error increase or decrease
as the sample size increases? Explain your answer.
9-74. Suppose that in Exercise 9-73, the experimenter had
believed that 14. For each of the following sample sizes,
and a fixed 0.05, find the probability of a type II error if
the true mean is 205 milliliters per hour.
(a)n 20
(b)n 50
(c)n 100
(d) Comparing your answers to those in Exercise 9-73, does
the probability of a type II error increase or decrease with
the increase in standard deviation? Explain your answer.
9-75. The marketers of shampoo products know that cus-
tomers like their product to have a lot of foam. A manufacturer
of shampoo claims that the foam height of his product exceeds
200 millimeters. It is known from prior experience that the
standard deviation of foam height is 8 millimeters. For each of
the following sample sizes, and a fixed 0.05, find the
power of the test if the true mean is 204 millimeters.
(a)n 20
(b)n 50
(c)n 100
(d) Does the power of the test increase or decrease as the sam-
ple size increases? Explain your answer.
9-76. Suppose we wish to test the hypothesis H 0 : 85
versus the alternative H 1 : 85 where 16. Suppose that
the true mean is 86 and that in the practical context of the
problem this is not a departure from 0 85 that has practical
significance.
(a) For a test with 0.01, compute for the sample sizes
n 25, 100, 400, and 2500 assuming that 86.
(b) Suppose the sample average is. Find the P-value
for the test statistic for the different sample sizes speci-
fied in part (a). Would the data be statistically significant
at 0.01?
(c) Comment on the use of a large sample size in this problem.
9-77. The cooling system in a nuclear submarine consists of
an assembly of welded pipes through which a coolant is circu-
lated. Specifications require that weld strength must meet or
exceed 150 psi.
(a) Suppose that the design engineers decide to test the
hypothesis H 0 : 150 versus H 1 : 150. Explainx 86why this choice of alternative hypothesis is better than
H 1 : 150.
(b) A random sample of 20 welds results in psi and
s 11.3 psi. What conclusions can you draw about the
hypothesis in part (a)? State any necessary assumptions
about the underlying distribution of the data.
9-78. Suppose we are testing H 0 : p 0.5 versus H 0 : p 0.5.
Suppose that pis the true value of the population proportion.
(a) Using 0.05, find the power of the test for n 100,
150, and 300 assuming that p 0.6. Comment on the
effect of sample size on the power of the test.
(b) Using 0.01, find the power of the test for n 100,
150, and 300 assuming that p 0.6. Compare your an-
swers to those from part (a) and comment on the effect of
on the power of the test for different sample sizes.
(c) Using 0.05, find the power of the test for n 100, as-
suming p 0.08. Compare your answer to part (a) and
comment on the effect of the true value of pon the power
of the test for the same sample size and level.
(d) Using 0.01, what sample size is required if p 0.6
and we want 0.05? What sample is required if
p 0.8 and we want 0.05? Compare the two sam-
ple sizes and comment on the effect of the true value of
pon sample size required when is held approximately
constant.
9-79. Consider the television picture tube brightness exper-
iment described in Exercise 8-24.
(a) For the sample size n 10, do the data support the
claim that the standard deviation of current is less than
20 microamps?
(b) Suppose instead of n 10, the sample size was 51.
Repeat the analysis performed in part (a) using n 51.
(c) Compare your answers and comment on how sample size
affects your conclusions drawn in parts (a) and (b).
9-80. Consider the fatty acid measurements for the diet
margarine described in Exercise 8-25.
(a) For the sample size n 6, using a two-sided alternative
hypothesis and 0.01, test H 0 : ^2 1.0.
(b) Suppose instead of n 6, the sample size was n 51.
Repeat the analysis performed in part (a) using n 51.
(c) Compare your answers and comment on how sample size
affects your conclusions drawn in parts (a) and (b).
9-81. A manufacturer of precision measuring instruments
claims that the standard deviation in the use of the instruments
is at most 0.00002 millimeter. An analyst, who is unaware of
the claim, uses the instrument eight times and obtains a sam-
ple standard deviation of 0.00001 millimeter.
(a) Confirm using a test procedure and an level of 0.01 that
there is insufficient evidence to support the claim that the
standard deviation of the instruments is at most 0.00002.
State any necessary assumptions about the underlying dis-
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