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288 CHAPTER 9 TESTS OF HYPOTHESES FOR A SINGLE SAMPLE

5. Determine an appropriate test statistic.
6. State the rejection region for the statistic.
7. Compute any necessary sample quantities, substitute these into the equation for the
test statistic, and compute that value.
8. Decide whether or not H 0 should be rejected and report that in the problem context.
Steps 1–4 should be completed prior to examination of the sample data. This sequence of
steps will be illustrated in subsequent sections.

EXERCISES FOR SECTION 9-1

9-1. In each of the following situations, state whether it is a test H 0 : 175 millimeters versus H 1 : 175 millime-
correctly stated hypothesis testing problem and why. ters, using the results of n 10 samples.

(a) H 0 : 25, H 1 : 25 (a) Find the type I error probability if the critical region is
(b) H 0 : 10, H 1 : 10 x 185 .
(c) H 0 : x 50, H 1 : x 50 (b) What is the probability of type II error if the true mean
(d) H 0 : p 0.1, H 1 : p 0.5 foam height is 195 millimeters?
(e) H 0 : s 30, H 1 : s 30 9-9. In Exercise 9-8, suppose that the sample data result in
9-2. A textile fiber manufacturer is investigating a new x 190 millimeters.
drapery yarn, which the company claims has a mean thread (a) What conclusion would you reach?
elongation of 12 kilograms with a standard deviation of 0.5 (b) How “unusual” is the sample value x 190 millimeters
kilograms. The company wishes to test the hypothesis if the true mean is really 175 millimeters? That is, what is
H 0 : 12 against H 1 : 12, using a random sample of the probability that you would observe a sample average
four specimens. as large as 190 millimeters (or larger), if the true mean
(a) What is the type I error probability if the critical region is foam height was really 175 millimeters?
defined as x 11.5 kilograms? 9-10. Repeat Exercise 9-8 assuming that the sample size is
(b) Find for the case where the true mean elongation is n 16 and the boundary of the critical region is the same.
11.25 kilograms.
9-11. Consider Exercise 9-8, and suppose that the sample
9-3. Repeat Exercise 9-2 using a sample size of n = 16 and size is increased to n 16.
the same critical region. (a) Where would the boundary of the critical region be placed
9-4. In Exercise 9-2, find the boundary of the critical region if the type I error probability were to remain equal to the
if the type I error probability is specified to be 0.01 . value that it took on when n 10?
9-5. In Exercise 9-2, find the boundary of the critical region (b) Using n 16 and the new critical region found in part (a),
if the type I error probability is specified to be 0.05. find the type II error probability if the true mean foam
9-6. The heat evolved in calories per gram of a cement mix- height is 195 millimeters.
ture is approximately normally distributed. The mean is (c) Compare the value of obtained in part (b) with the value
thought to be 100 and the standard deviation is 2. We wish to from Exercise 9-8 (b). What conclusions can you draw?
test H 0 : 100 versus H 1 : 100 with a sample of 9-12. A manufacturer is interested in the output voltage of a
n = 9 specimens. power supply used in a PC. Output voltage is assumed to be
(a) If the acceptance region is defined as 98.5 x 101.5 , normally distributed, with standard deviation 0.25 Volts, and
find the type I error probability . the manufacturer wishes to test H 0 : 5 Volts against
(b) Find for the case where the true mean heat evolved is 103. H 1 : 5 Volts, using n 8 units.
(c) Find for the case where the true mean heat evolved is (a) The acceptance region is 4.85 x 5.15. Find the value
105. This value of is smaller than the one found in part of .
(b) above. Why? (b) Find the power of the test for detecting a true mean output
9-7. Repeat Exercise 9-6 using a sample size of n 5 and voltage of 5.1 Volts.
the same acceptance region. 9-13. Rework Exercise 9-12 when the sample size is 16 and
9-8. A consumer products company is formulating a new the boundaries of the acceptance region do not change.
shampoo and is interested in foam height (in millimeters). 9-14. Consider Exercise 9-12, and suppose that the manu-
Foam height is approximately normally distributed and has a facturer wants the type I error probability for the test to be
standard deviation of 20 millimeters. The company wishes to 0.05. Where should the acceptance region be located?