Answer
a) The null hypothesis can be rejected at $\alpha =0.10$ and this means that the owners are not having their transmissions serviced at 30,000 miles.
b) I think the $\alpha$ value of 0.10 is an appropriate significance level.
Work Step by Step
Given that, $\mu=30000, \sigma=1684, n=40, \bar x =30456$, then,
$H_{0}:\mu=30000.\\ H_{1}:\mu \ne 30000.$
$Z=\frac{\bar x- \mu}{\frac{\sigma}{\sqrt{n}}}=\frac{30456- 30000}{\frac{1684}{\sqrt{40}}}\approx 1.71$
$P-value =2\times(1-P(Z\leq 1.71)=2\times(1-0.9564)=0.0872$
Since $\alpha=0.10$ and $P-value =0.0872$, the null hypothesis can be rejected.
