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324 CHAPTER 9 TESTS OF HYPOTHESES FOR A SINGLE SAMPLE

9-73. An inspector of flow metering devices used to admin- why this choice of alternative hypothesis is better than
ister fluid intravenously will perform a hypothesis test to H 1 : 150.
determine whether the mean flow rate is different from the flow (b) A random sample of 20 welds results in x 153.7 psi and
rate setting of 200 milliliters per hour. Based on prior s 11.3 psi. What conclusions can you draw about the
information the standard deviation of the flow rate is assumed hypothesis in part (a)? State any necessary assumptions
to be known and equal to 12 milliliters per hour. For each of the about the underlying distribution of the data.
following sample sizes, and a fixed 0.05, find the probabil- 9-78. Suppose we are testing H 0 : p 0.5 versus H 0 : p 0.5.
ity of a type II error if the true mean is 205 milliliters per hour. Suppose that p is the true value of the population proportion.
(a) n 20 (a) Using 0.05, find the power of the test for n 100,
(b) n 50 150, and 300 assuming that p 0.6. Comment on the
(c) n 100 effect of sample size on the power of the test.
(d) Does the probability of a type II error increase or decrease (b) Using 0.01, find the power of the test for n 100,
as the sample size increases? Explain your answer. 150, and 300 assuming that p 0.6. Compare your an-
9-74. Suppose that in Exercise 9-73, the experimenter had swers to those from part (a) and comment on the effect of
believed that 14. For each of the following sample sizes, on the power of the test for different sample sizes.
and a fixed 0.05, find the probability of a type II error if (c) Using 0.05, find the power of the test for n 100, as-
the true mean is 205 milliliters per hour. suming p 0.08. Compare your answer to part (a) and
(a) n 20 comment on the effect of the true value of p on the power
(b) n 50 of the test for the same sample size and level.
(c) n 100 (d) Using 0.01, what sample size is required if p 0.6
(d) Comparing your answers to those in Exercise 9-73, does and we want 0.05? What sample is required if
the probability of a type II error increase or decrease with p 0.8 and we want 0.05? Compare the two sam-
the increase in standard deviation? Explain your answer. ple sizes and comment on the effect of the true value of
9-75. The marketers of shampoo products know that cus- p on sample size required when is held approximately
tomers like their product to have a lot of foam. A manufacturer constant.
of shampoo claims that the foam height of his product exceeds 9-79. Consider the television picture tube brightness exper-
200 millimeters. It is known from prior experience that the iment described in Exercise 8-24.
standard deviation of foam height is 8 millimeters. For each of (a) For the sample size n 10, do the data support the
the following sample sizes, and a fixed 0.05, find the claim that the standard deviation of current is less than
power of the test if the true mean is 204 millimeters. 20 microamps?
(a) n 20 (b) Suppose instead of n 10, the sample size was 51.
(b) n 50 Repeat the analysis performed in part (a) using n 51.
(c) n 100 (c) Compare your answers and comment on how sample size
(d) Does the power of the test increase or decrease as the sam- affects your conclusions drawn in parts (a) and (b).
ple size increases? Explain your answer. 9-80. Consider the fatty acid measurements for the diet
9-76. Suppose we wish to test the hypothesis H 0 : 85 margarine described in Exercise 8-25.
versus the alternative H 1 : 85 where 16. Suppose that (a) For the sample size n 6, using a two-sided alternative
2
the true mean is 86 and that in the practical context of the hypothesis and 0.01, test H 0 : 1.0.
problem this is not a departure from 0 85 that has practical (b) Suppose instead of n 6, the sample size was n 51.
significance. Repeat the analysis performed in part (a) using n 51.
(a) For a test with 0.01, compute for the sample sizes (c) Compare your answers and comment on how sample size
n 25, 100, 400, and 2500 assuming that 86. affects your conclusions drawn in parts (a) and (b).
(b) Suppose the sample average is x 86 . Find the P-value 9-81. A manufacturer of precision measuring instruments
for the test statistic for the different sample sizes speci- claims that the standard deviation in the use of the instruments
fied in part (a). Would the data be statistically significant is at most 0.00002 millimeter. An analyst, who is unaware of
at 0.01? the claim, uses the instrument eight times and obtains a sam-
(c) Comment on the use of a large sample size in this problem. ple standard deviation of 0.00001 millimeter.
9-77. The cooling system in a nuclear submarine consists of (a) Confirm using a test procedure and an level of 0.01 that
an assembly of welded pipes through which a coolant is circu- there is insufficient evidence to support the claim that the
lated. Specifications require that weld strength must meet or standard deviation of the instruments is at most 0.00002.
exceed 150 psi. State any necessary assumptions about the underlying dis-
(a) Suppose that the design engineers decide to test the tribution of the data.
hypothesis H 0 : 150 versus H 1 : 150. Explain