Question 8.7: Can we conclude from the Hooke’s law data that the unloaded ......
Can we conclude from the Hooke’s law data that the unloaded length of the spring is more than 4.9 in.?
This requires a hypothesis test. The null and alternate hypotheses are
H_{0} : \beta_{0} ≤ 4.9 versus H_{1} : \beta_{0} > 4.9
The quantity
\frac{\hat{\beta}_{0}\ −\ \beta_{0}}{s_{\hat{\beta}_{0}}}
has a Student’s t distribution with n − 2 = 20 − 2 = 18 degrees of freedom. Under H_{0}, we take \beta_{0} = 4.9. The test statistic is therefore
\frac{\hat{\beta}_{0}\ −\ 4.9}{s_{\hat{\beta}_{0}}}
We have previously computed \hat{\beta}_{0} = 4.9997 and s_{\hat{\beta }{_{0}}} = 0.0248. The value of the test statistic is therefore
\frac{4.9997\ −\ 4.9}{0.0248} = 4.020
Consulting the Student’s t table, we find that the P-value is less than 0.0005. (Software yields P = 0.0004.) We can conclude that the unloaded length of the spring is more than 4.9 in.