10-6 INFERENCE ON TWO POPULATION PROPORTIONS 36110-6 INFERENCE ON TWO POPULATION PROPORTIONSWe now consider the case where there are two binomial parameters of interest, say, p 1 and p 2 ,
and we wish to draw inferences about these proportions. We will present large-sample
hypothesis testing and confidence interval procedures based on the normal approximation to
the binomial.10-6.1 Large-Sample Test for H 0 : p 1 p 2Suppose that two independent random samples of sizes n 1 and n 2 are taken from two pop-
ulations, and let X 1 and X 2 represent the number of observations that belong to the class of in-
terest in samples 1 and 2, respectively. Furthermore, suppose that the normal approximation
to the binomial is applied to each population, so the estimators of the population proportionsdiffer between the two companies. The standard deviation of
concentration in a random sample of n 1 10 batches pro-
duced by company 1 is s 1 4.7 grams per liter, while for
company 2, a random sample of n 2 16 batches yields s 2
5.8 grams per liter. Is there sufficient evidence to conclude
that the two population variances differ? Use 0.05.
10-48. Consider the etch rate data in Exercise 10-21. Test
the hypothesis H 0 : ^21 ^22 against H 1 : ^21 ^22 using
0.05, and draw conclusions.
10-49. Consider the etch rate data in Exercise 10-21.
Suppose that if one population variance is twice as large as the
other, we want to detect this with probability at least 0.90
(using 0.05). Are the sample sizes n 1 n 2 10 adequate?
10-50. Consider the diameter data in Exercise 10-17. Con-
struct the following:
(a) A 90% two-sided confidence interval on 1 2.
(b) A 95% two-sided confidence interval on 1 2. Comment
on the comparison of the width of this interval with the
width of the interval in part (a).
(c) A 90% lower-confidence bound on 1 2.
10-51. Consider the foam data in Exercise 10-18. Construct
the following:
(a) A 90% two-sided confidence interval on ^21 ^22.
(b) A 95% two-sided confidence interval on ^21 ^22. Comment
on the comparison of the width of this interval with the
width of the interval in part (a).
(c) A 90% lower-confidence bound on 1 2.
10-52. Consider the film speed data in Exercise 10-24. Test
H 0 : ^21 ^22 versus using 0.02.
10-53. Consider the gear impact strength data in Exercise
10-22. Is there sufficient evidence to conclude that the vari-
ance of impact strength is different for the two suppliers?
Use 0.05.
10-54. Consider the melting point data in Exercise 10-25.
Do the sample data support a claim that both alloys have theH 1 : ^21 ^22same variance of melting point? Use 0.05 in reaching
your conclusion.
10-55. Exercise 10-28 presented measurements of plastic
coating thickness at two different application temperatures.
Test H 0 : ^21 ^22 against using 0.01.
10-56. A study was performed to determine whether men
and women differ in their repeatability in assembling compo-
nents on printed circuit boards. Random samples of 25 men
and 21 women were selected, and each subject assembled the
units. The two sample standard deviations of assembly time
were smen0.98 minutes and swomen1.02 minutes. Is there
evidence to support the claim that men and women differ in
repeatability for this assembly task? Use 0.02 and state
any necessary assumptions about the underlying distribution
of the data.
10-57. Reconsider the assembly repeatability experiment
described in Exercise 10-56. Find a 98% confidence interval
on the ratio of the two variances. Provide an interpretation of
the interval.
10-58. Reconsider the film speed experiment in Exercise
10-24. Suppose that one population standard deviation is 50%
larger than the other. Is the sample size n 1 n 2 8 adequate
to detect this difference with high probability? Use 0.01
in answering this question.
10-59. Reconsider the overall distance data for golf balls in
Exercise 10-31. Is there evidence to support the claim that the
standard deviation of overall distance is the same for both
brands of balls (use 0.05)? Explain how this question can
be answered with a 95% confidence interval on.
10-60. Reconsider the coefficient of restitution data in
Exercise 10-32. Do the data suggest that the standard devia-
tion is the same for both brands of drivers (use 0.05)?
Explain how to answer this question with a confidence inter-
val on . 1
2 1
2
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