Exercise 54 in Chapter 7 presented a t variable appropriate for making inferences about _{1} _{2} when both population distributions are normal and, in addition, it can be assumed that _{1} _{2}.

a. Describe how this variable can be used to form a test statistic and test procedure, the pooled t test, for testing H_{0}: _{1} _{2} .

b. Use the pooled t test to test the relevant hypotheses based on the SAS output given in Exercise 33.

c. Use the pooled t test to reach a conclusion in Exercise 35.

Exercise 54

Suppose not only that the two population or treatment response distributions are normal but also that they have equal variances. Let ^{2} denote the common variance. This variance can be estimated by a “pooled” (i.e., combined) sample variance as follows:

(n_{1} n_{2} 2 is the sum of the df’s contributed by the two samples). It can then be shown that the standardized variable

has a t distribution with n_{1} n_{2} 2 df.

a. Use the t variable above to obtain a pooled t confidence interval formula for _{1} _{2}.

b. A sample of ultrasonic humidifiers of one particular brand was selected for which the observations on maximum output of moisture (oz) in a controlled chamber were 14.0, 14.3, 12.2, and 15.1. A sample of the second brand gave output values 12.1, 13.6, 11.9, and 11.2 (“Multiple Comparisons of Means Using Simultaneous Confidence Intervals,” J. of Quality Technology, 1989: 232–241). Use the pooled t formula from part (a) to estimate the difference between true average outputs for the two brands with a 95% confidence interval.

c. Estimate the difference between the two s using the two-sample t interval discussed in this section, and compare it to the interval of part (b).