A fire insurance company felt 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 felt, would convince the insurance company to lower its rates. They randomly identify 64 homes and measured the distance to the nearest fire department for 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 α = 0.05 level of significance?
What would my null and alternate hypothesis be?
To determine the null and alternative hypotheses for this hypothesis test, we need to consider the claim made by the community and the values given in the problem.
Null hypothesis (H₀): The mean distance from a home to the nearest fire department in the suburb of Chicago is equal to or greater than 4.7 miles.
Alternate hypothesis (H₁): The mean distance from a home to the nearest fire department in the suburb of Chicago is less than 4.7 miles.
In equation form:
H₀: μ ≥ 4.7
H₁: μ < 4.7
In this case, the null hypothesis assumes that the insurance company's claim is correct, and the alternate hypothesis challenges that claim by asserting that the mean distance is in fact less than 4.7 miles.
The null hypothesis (H0): The mean distance from a home to the nearest fire department is 4.7 miles.
The alternative hypothesis (H1): The mean distance from a home to the nearest fire department is less than 4.7 miles.