Chapter 9.2, Problem 34E

Consider the pooled t variable

$T=\frac{(\bar{X}-\bar{Y})-({\mu }_1-{\mu }_2)}{sp\sqrt{\frac{1}{m}}+\frac{1}{n}}$

which has a t distribution with m + n − 2 df when both population distributions are normal with σ₁ = σ₂ (see the Pooled t Procedures subsection for a description of S_p).

a. Use this t variable to obtain a pooled t confidence interval formula for μ₁ − μ₂.

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 μ’s using the two-sample t interval discussed in this section, and compare it to the interval of part (b).