Answer:
a) Null and alternative hypothesis
![H_0: \mu=1503\\\\H_a:\mu< 1503](https://tex.z-dn.net/?f=H_0%3A%20%5Cmu%3D1503%5C%5C%5C%5CH_a%3A%5Cmu%3C%201503)
b) Point estimate d = -$78
c) Test statistic t = -2.438
P-value = 0.0113
Reject H0. We can conclude that the population mean automobile premium in Pennsylvania is lower than the national mean.
Step-by-step explanation:
This is a hypothesis test for the population mean.
The claim is that automobile insurance in Pennsylvania is significantly cheaper than the national average.
Then, the null and alternative hypothesis are:
The significance level is 0.05.
The sample has a size n=25.
The sample mean is M=1425.
A point estimate of the difference between the mean annual premium in Pennsylvania and the national mean can be calculated with the sample mean:
![d=M-\mu=1425-1503=-78](https://tex.z-dn.net/?f=d%3DM-%5Cmu%3D1425-1503%3D-78)
As the standard deviation of the population is not known, we estimate it with the sample standard deviation, that has a value of s=160.
The estimated standard error of the mean is computed using the formula:
Then, we can calculate the t-statistic as:
![t=\dfrac{M-\mu}{s/\sqrt{n}}=\dfrac{1425-1503}{32}=\dfrac{-78}{32}=-2.438](https://tex.z-dn.net/?f=t%3D%5Cdfrac%7BM-%5Cmu%7D%7Bs%2F%5Csqrt%7Bn%7D%7D%3D%5Cdfrac%7B1425-1503%7D%7B32%7D%3D%5Cdfrac%7B-78%7D%7B32%7D%3D-2.438)
The degrees of freedom for this sample size are:
This test is a left-tailed test, with 24 degrees of freedom and t=-2.438, so the P-value for this test is calculated as (using a t-table):
![\text{P-value}=P(t](https://tex.z-dn.net/?f=%5Ctext%7BP-value%7D%3DP%28t%3C-2.438%29%3D0.0113)
As the P-value (0.0113) is smaller than the significance level (0.05), the effect is significant.
The null hypothesis is rejected.
At a significance level of 0.05, there is enough evidence to support the claim that automobile insurance in Pennsylvania is significantly cheaper than the national average.