Question

The mean annual premium for automobile insurance in the United States is $1,499. Being from Pennsylvania, you believe automobile insurance is cheaper there and wish 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 $1,440 with a standard deviation of s = $150.

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.
b. What is a point estimate of the difference between the mean annual premium in Pennsylvania and the national mean (to the nearest dollar)?

Answer 1

a) Then, t(s) is in the rejection region. We can not support that the mean annual premium in Pennsylvania is lower than the national mean annual premium

b) The point estimate of the difference is 59 $

Explanation:

From sample we get:

sample size n = 25

annual mean x = 1440

standard deviation of the sample s = 150

mean for the mean annual premium μ = 1499

Hypothesis Test:

Null Hypothesis H₀ x = μ

Alternative Hypothesis Hₐ x < μ

Alternative Hypotesis indicates that the test is a one-tail test to the left

Let´s assume confidence Interval 95 % then α = 5 % α = 0.05

As n < 30 and we assume normality we will make a t-student test

degree of freedom df = n - 1 df = 24

With df = 24 and α = 0.05 and from t-student table we get t(c) = - 1.711

To calculate t(s) we apply:

t(s) = ( x - μ ) / s/√n

t(s) = ( 1440 - 1499 ) * 5 / 150

t(s) = - 59/ 30

t(s) = - 1.96

Comparing t(s) and t(c)

|t(s)| > |t(c)|

Then, t(s) is in the rejection region. We can not support that the mean annual premium in Pennsylvania is lower than the national mean annual premium.

b) The point estimate for the difference is : x - μ = 59 $