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336 CHAPTER 10 STATISTICAL INFERENCE FOR TWO SAMPLES

and a 100(1 )% lower-confidence bound is

2 2
1 2
x x z 2 (10-10)
1
1

2
n
B 1 n 2
EXERCISES FOR SECTION 10-2
10-1. Two machines are used for filling plastic bottles with and n 2 20 specimens are tested; the sample mean burn-
a net volume of 16.0 ounces. The fill volume can be assumed ing rates are x 1 18 centimeters per second and x 2 24
normal, with standard deviation 1 0.020 and 2 0.025 centimeters per second.
ounces. A member of the quality engineering staff suspects (a) Test the hypothesis that both propellants have the same
that both machines fill to the same mean net volume, whether mean burning rate. Use 0.05.
or not this volume is 16.0 ounces. A random sample of 10 bot- (b) What is the P-value of the test in part (a)?
tles is taken from the output of each machine. (c) What is the -error of the test in part (a) if the true differ-
ence in mean burning rate is 2.5 centimeters per second?
Machine 1 Machine 2 (d) Construct a 95% confidence interval on the difference in
16.03 16.01 16.02 16.03 means 1 2 . What is the practical meaning of this
interval?
16.04 15.96 15.97 16.04
10-5. Two machines are used to fill plastic bottles with
16.05 15.98 15.96 16.02
dishwashing detergent. The standard deviations of fill volume
16.05 16.02 16.01 16.01
are known to be 1 0.10 fluid ounces and 2 0.15 fluid
16.02 15.99 15.99 16.00 ounces for the two machines, respectively. Two random sam-
ples of n 1 12 bottles from machine 1 and n 2 10 bottles
(a) Do you think the engineer is correct? Use 0.05. from machine 2 are selected, and the sample mean fill vol-
(b) What is the P-value for this test?
umes are x 1 30.87 fluid ounces and x 2 30.68 fluid
(c) What is the power of the test in part (a) for a true differ- ounces. Assume normality.
ence in means of 0.04? (a) Construct a 90% two-sided confidence interval on the
(d) Find a 95% confidence interval on the difference in mean difference in fill volume. Interpret this interval.
means. Provide a practical interpretation of this interval. (b) Construct a 95% two-sided confidence interval on the mean
(e) Assuming equal sample sizes, what sample size should be difference in fill volume. Compare and comment on the
used to assure that 0.05 if the true difference in width of this interval to the width of the interval in part (a).
means is 0.04? Assume that 0.05. (c) Construct a 95% upper-confidence interval on the mean
10-2. Two types of plastic are suitable for use by an elec- difference in fill volume. Interpret this interval.
tronics component manufacturer. The breaking strength of this 10-6. Reconsider the situation described in Exercise 10-5.
plastic is important. It is known that 1 2 1.0 psi. From (a) Test the hypothesis that both machines fill to the same
a random sample of size n 1 10 and n 2 12, we obtain mean volume. Use 0.05.
x 1 162.5 and x 2 155.0 . The company will not adopt plas- (b) What is the P-value of the test in part (a)?
tic 1 unless its mean breaking strength exceeds that of plastic (c) If the -error of the test when the true difference in fill
2 by at least 10 psi. Based on the sample information, should volume is 0.2 fluid ounces should not exceed 0.1, what
it use plastic 1? Use 0.05 in reaching a decision. sample sizes must be used? Use 0.05.
10-3. Reconsider the situation in Exercise 10-2. Suppose 10-7. Two different formulations of an oxygenated motor fuel
that the true difference in means is really 12 psi. Find the are being tested to study their road octane numbers. The variance
power of the test assuming that 0.05. If it is really impor- 2
of road octane number for formulation 1 is 1 1.5, and for
tant to detect this difference, are the sample sizes employed in formulation 2 it is 2 1.2. Two random samples of size n 1 15
2
Exercise 10-2 adequate, in your opinion? and n 2 20 are tested, and the mean road octane numbers
10-4. The burning rates of two different solid-fuel propel- observed are x 1 89.6 and x 2 92.5. Assume normality.
lants used in aircrew escape systems are being studied. It is (a) Construct a 95% two-sided confidence interval on the
known that both propellants have approximately the same difference in mean road octane number.
standard deviation of burning rate; that is 1 2 3 (b) If formulation 2 produces a higher road octane number
centimeters per second. Two random samples of n 1 20 than formulation 1, the manufacturer would like to detect