10-7 SUMMARY TABLE FOR INFERENCE PROCEDURES FOR TWO SAMPLES 371IMPORTANT TERMS AND CONCEPTS
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Comparative experi-
ments
Critical region for a test
statistic
Identifying cause and
effectNull and alternative
hypotheses
One-sided and two-
sided alternative
hypotheses
Operating characteristic
curves
Paired t-test
Pooled t-testP-value
Reference distribution
for a test statistic
Sample size determina-
tion for hypothesis
tests and confidence
intervals
Statistical hypotheses
Test statisticCD MATERIAL
Fisher-Irwin test on two
proportionsMIND-EXPANDING EXERCISES(b) Do these data support the claim that pesticide 3 is
more effective than the other two? Use 0.05 in
determining your answer.
10-91. Suppose that we wish to test H 0 : 1 2
versus H 1 : 1 2 , where 12 and ^22 are known. The
total sample size Nis to be determined, and the alloca-
tion of observations to the two populations such that
n 1 n 2 Nis to be made on the basis of cost. If the
cost of sampling for populations 1 and 2 are C 1 and C 2 ,
respectively, find the minimum cost sample sizes that
provide a specified variance for the difference in sam-
ple means.
10-92. Suppose that we wish to test the hypothesis H 0 :
1 2 versus H 1 : 1 2 , where both variances 12 and
^22 are known. A total of n 1 n 2 Nobservations can be
taken. How should these observations be allocated to the
two populations to maximize the probability that H 0 will
be rejected if H 1 is true and 1 2 0?
10-93. Suppose that we wish to test H 0 : 0 ver-
sus H 1 : 0 , where the population is normal with
known . Let 0 , and define the critical region
so that we will reject H 0 if z 0
zor if z 0 z ,
where z 0 is the value of the usual test statistic for these
hypotheses.(a) Show that the probability of type I error for this test
is.
(b) Suppose that the true mean is 1 0 . Derive
an expression for for the above test.
10-94. Construct a data set for which the paired t-test
statistic is very large, indicating that when this analysis
is used the two population means are different, but t 0 for
the two-sample t-test is very small so that the incorrect
analysis would indicate that there is no significant dif-
ference between the means.
10-95. In some situations involving proportions, we
are interested in the ratio p 1 p 2 rather than the differ-
ence p 1 p 2. Let . We can show that ln( ) has
an approximate normal distribution with the mean (n)
and variance
(a) Use the information above to derive a large-sample
confidence interval for ln .
(b) Show how to find a large-sample CI for .
(c) Use the data from the St. John’s Wort study in
Example 10-14, and find a 95% CI on p 1 p 2.
Provide a practical interpretation for this CI.
10-96. Derive an expression for for the test of the
equality of the variances of two normal distributions.
Assume that the two-sided alternative is specified.31 n 1 x 12
1 n 1 x 12 1 n 2 x 22
1 n 2 x 2241
2.
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