Question

A property and casualty insurance company (which provides fire coverage for dwellings) felt that the mean distance from a home to the nearest fire department in rural Alabama was at least 10 miles. It set its fire insurance rates accordingly. Members of the community set out to show that the mean distance was less than 10 miles due to the increased number of volunteer fire departments. This, they felt, would convince the insurance company to lower its rates. They randomly identify 64 homes and measure the distance to the nearest fire department for each. The resulting sample mean was 8.7 miles. If σ = 3.5 miles, does the sample show sufficient evidence to support the community’s claim? Use the four step process for Hypothesis Testing.

Step 1 – State Hypothesis in context of the problem.

Step 2 – Gather data, check assumptions, and find rejection region using α.

Step 3 – Calculate the appropriate test statistic and p-value.

Step 4 – State conclusion in context of the problem.

Answer 1

STEP 1

Community members want to test whether the mean distance is less than 10 miles or not. So, it is a left tailed hypothesis test.

Null hypothesis

Alternate hypothesis

STEP 2

We have

Sample size is greater than 30 and population standard deviation is given, So we will use z test for the hypothesis testing.

Significance level for testing is 0.05

Reject null hypothesis when p value is less than 0.05

STEP 3

Calculation for the test statistic

z =

setting the given values, we get

z =

Now, we have to find the probability corresponding to the calculated z statistic

using the identity

we can write

using z distribution table, we get

= 1 - 0.9985 = 0.0015

So, the calculated p value is 0.0015 which is less than 0.05 level of significance.

STEP 4

P value is significant, we will reject the null hypothesis

thus, we can conclude that we have sufficient evidence to support the claim that the mean distance was less than 10 miles due to the increased number of volunteer fire departments.