Solved: Reconsider the situation of Exercise 73, in which

Chapter 12, Problem 87E

(choose chapter or problem)

Reconsider the situation of Exercise 73, in which x = retained austenite content using a garnet abrasive and y = abrasive wear loss were related via the simple linear regression model\(Y=\beta_{0}+\beta_{1} x+\varepsilon\). Suppose that for a second type of abrasive, these variables are also related via the simple linear regression model \(Y=\gamma_{0}+\gamma_{1} x+\varepsilon\) and that \(V(\boldsymbol{\epsilon})=\sigma^{2}\) for both types of abrasive. If the data set consists of n1 observations on the first abrasive and n2 on the second and if SSE1 and SSE2 denote the two error sums of squares, then a pooled estimate of σ2 is \(\hat{\sigma}^{2}=\left(\mathrm{SSE}_{1}+\mathrm{SSE}_{2}\right) /\left(n_{1}+n_{2}-4\right)\) denote \(\sum\left(x_{i}-\bar{x}\right)^{2}\) or the data on the first and second abrasives, respectively. A test of \(H_{0}: \beta_{1}-\gamma_{1}=0\) (equal slopes) is based on the statistic

\(T=\frac{\hat{\beta}_{1}-\hat{\gamma}_{1}}{\hat{\sigma} \sqrt{\frac{1}{\mathrm{SS}_{x 1}}+\frac{1}{\mathrm{SS}_{x 2}}}}\)

When H0 is true, T has a t distribution with n1 + n2 - 4 df. Suppose the 15 observations using the alternative abrasive Give \(S S_{x 2}=7152.5578, \hat{\gamma}_{1}=.006845 . \text { and } \mathrm{SSE}_{2}=.51350\), and Using this along with the data of Exercise 73, carry out a test at level .05 to see whether expected change in wear loss associated with a 1% increase in austenite content is identical for the two types of abrasive.