Question 8.6: The manufacturer of the spring in the Hooke’s law data claim......
The manufacturer of the spring in the Hooke’s law data claims that the spring constant \beta_{1} is at least 0.215 in./lb. We have estimated the spring constant to be \hat{\beta}_{1} = 0.2046 in./lb. Can we conclude that the manufacturer’s claim is false?
This calls for a hypothesis test. The null and alternate hypotheses are
H_{0} : \beta_{1} ≥ 0.215 versus H_{1} : \beta_{1} < 0.215
The quantity
\frac{\hat{\beta}_{1}\ −\ \beta _{1}}{s_{\hat{\beta}_{1}}}
has a Student’s t distribution with n − 2 = 20 − 2 = 18 degrees of freedom. Under H_{0}, we take \beta_{1} = 0.215. The test statistic is therefore
\frac{\hat{\beta}_{1}\ −\ 0.215}{s_{\hat{\beta}_{1}}}
We have previously computed \hat{\beta}_{1} = 0.2046 and s_{\hat{\beta}_{1}} = 0.0111. The value of the test statistic is therefore
\frac{0.2046\ −\ 0.215}{0.0111} = −0.937
Consulting the Student’s t table, we find that the P-value is between 0.10 and 0.25. (Software yields P = 0.181.) We cannot reject the manufacturer’s claim on the basis of these data.