The Insure.com website reports that the mean annual premium for automobile insurance in the United States was $1,503 in March 20

Question

2 months ago

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The Insure.com website reports that the mean annual premium for automobile insurance in the United States was $1,503 in March 20

14. Being from Pennsylvania at that time, you believed automobile insurance was cheaper there and decided to develop statistical support for your opinion. A sample of 25 automobile insurance policies from the state of Pennsylvania showed a mean annual premium of $1,425 with a standard deviation of
s = $160.
(a) Develop a hypothesis test that can be used to determine whether the mean annual premium in Pennsylvania is lower than the national mean annual premium.
H0: μ ≥ 1,503
Ha: μ < 1,503
H0: μ ≤ 1,503
Ha: μ > 1,503
H0: μ > 1,503
Ha: μ ≤ 1,503
H0: μ < 1,503
Ha: μ ≥ 1,503
H0: μ = 1,503
Ha: μ ≠ 1,503
(b) What is a point estimate in dollars of the difference between the mean annual premium in Pennsylvania and the national mean? (Use the mean annual premium in Pennsylvania minus the national mean.)
$
(c) At
α = 0.05,
test for a significant difference.
Find the value of the test statistic. (Round your answer to three decimal places.)
Find the p-value. (Round your answer to four decimal places.)
p-value =
State your conclusion.
Do not reject H0. We can conclude that the population mean automobile premium in Pennsylvania is lower than the national mean.Reject H0. We cannot conclude that the population mean automobile premium in Pennsylvania is lower than the national mean. Do not reject H0. We cannot conclude that the population mean automobile premium in Pennsylvania is lower than the national mean.Reject H0. We can conclude that the population mean automobile premium in Pennsylvania is lower than the national mean.

Mathematics

1 answer:

PIT_PIT [12.4K] · Answer 1 · 2026-03-18T10:12:52+03:00

Answer:

a) Null and alternative hypothesis

$H_0: \mu=1503\\\\H_a:\mu< 1503$

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:

$H_0: \mu=1503\\\\H_a:\mu< 1503$

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:

$d=M-\mu=1425-1503=-78$

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:

$s_M=\dfrac{s}{\sqrt{n}}=\dfrac{160}{\sqrt{25}}=32$

Then, we can calculate the t-statistic as:

$t=\dfrac{M-\mu}{s/\sqrt{n}}=\dfrac{1425-1503}{32}=\dfrac{-78}{32}=-2.438$

The degrees of freedom for this sample size stand at:

$df=n-1=25-1=24$

This constitutes a left-tailed test, with 24 degrees of freedom and t=-2.438, rendering the P-value as (per a t-table):

$\text{P-value}=P(t$

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.