Answer:
a) Null and alternative hypothesis
b) Point estimate d = -$78
c) Test statistic t = -2.438
P-value = 0.0113
Reject H0. This indicates that the average automobile premium in Pennsylvania is lower than in the nation.
Step-by-step breakdown:
This is a statistical test for the average population mean.
The hypothesis posits that car insurance in Pennsylvania is notably less expensive compared to the national average.
Accordingly, the null and alternative hypotheses are:
The significance level is set at 0.05.
The sample size is n=25.
The sample mean equates to M=1425.
A point estimate of the difference between the Pennsylvania mean premium and the national average can be computed using the sample mean:
Given that the standard deviation of the population is unknown, we approximate it using the sample standard deviation, which is s=160.
The estimated standard error of the mean is determined with the formula:
Then, we can calculate the t-statistic as:
The degrees of freedom for this sample size stand at:
This constitutes a left-tailed test, with 24 degrees of freedom and t=-2.438, rendering the P-value as (per a t-table):
As the P-value (0.0113) falls below the significance level (0.05), the results prove significant.
Thus, the null hypothesis obtains dismissal.
At a 0.05 significance level, there's sufficient evidence to assert that car insurance in Pennsylvania costs notably less than the national average.