Question

A fire insurance company thought that the mean distance from a home to the nearest fire department in a suburb of Chicago was at least 4.7 miles. It set its fire insurance rates accordingly. Members of the community set out to show that the mean distance was less than 4.7 miles. This, they thought, would convince the insurance company to lower its rates. They randomly identified 64 homes and measured the distance to the nearest fire department from each. The resulting sample mean was 4.4. If σ = 2.4 miles, does the sample show sufficient evidence to support the community's claim at the α = 0.05 level of significance?

A.) Conduct a hypothesis test using the classical approach.

B.) Conduct a hypothesis test using the p-value approach.

Answer 1

a)

Below are the null and alternative Hypothesis,
Null Hypothesis, H0: μ = 4.7
Alternative Hypothesis, Ha: μ < 4.7

Rejection Region
This is left tailed test, for α = 0.05
Critical value of z is -1.645.
Hence reject H0 if z < -1.645

Test statistic,
z = (xbar - mu)/(sigma/sqrt(n))
z = (4.4 - 4.7)/(2.4/sqrt(64))
z = -1

fail to reject null hypothesis.

b)

Below are the null and alternative Hypothesis,
Null Hypothesis, H0: μ = 4.7
Alternative Hypothesis, Ha: μ < 4.7

Test statistic,
z = (xbar - mu)/(sigma/sqrt(n))
z = (4.4 - 4.7)/(2.4/sqrt(64))
z = -1

P-value Approach
P-value = 0.1587
As P-value >= 0.05, fail to reject null hypothesis.