Respuesta :
Answer:
a) Null and alternative hypothesis
[tex]H_0: \mu=1503\\\\H_a:\mu< 1503[/tex]
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:
[tex]H_0: \mu=1503\\\\H_a:\mu< 1503[/tex]
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:
[tex]d=M-\mu=1425-1503=-78[/tex]
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:
[tex]s_M=\dfrac{s}{\sqrt{n}}=\dfrac{160}{\sqrt{25}}=32[/tex]
Then, we can calculate the t-statistic as:
[tex]t=\dfrac{M-\mu}{s/\sqrt{n}}=\dfrac{1425-1503}{32}=\dfrac{-78}{32}=-2.438[/tex]
The degrees of freedom for this sample size are:
[tex]df=n-1=25-1=24[/tex]
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):
[tex]\text{P-value}=P(t<-2.438)=0.0113[/tex]
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.
