What would the null and alternate hypothesis be?
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
Same as previous post.
To determine the null and alternate hypotheses for this scenario, we need to consider the goal of the community and the information provided.
Null hypothesis (H0): The mean distance from a home to the nearest fire department is at least 4.7 miles.
Alternate hypothesis (Ha): The mean distance from a home to the nearest fire department is less than 4.7 miles.
In this case, the null hypothesis represents the initial claim made by the fire insurance company, stating that the mean distance from a home to the nearest fire department is at least 4.7 miles. The alternate hypothesis represents the claim made by the community members, who believe that the mean distance is less than 4.7 miles.
To determine which hypothesis we should support based on the sample data, we need to conduct a hypothesis test using the sample mean distance and the known standard deviation (σ) provided.
To perform the hypothesis test and obtain a conclusion, we typically calculate a test statistic and compare it to a critical value or p-value. However, since the sample size is large (n = 64) and the population standard deviation (σ) is known, we can use a z-test.
The z-test compares the test statistic (calculated using the sample mean, population mean, and known standard deviation) to the critical value (obtained from the standard normal distribution) to determine the probability of obtaining the observed result.
In this case, the test statistic (z-score) can be calculated as:
z = (sample mean - population mean) / (population standard deviation / √n)
Using the given information:
Sample mean (x̄) = 4.4 miles
Population mean (μ) = 4.7 miles
Population standard deviation (σ) = 2.4 miles
Sample size (n) = 64 homes
Substituting these values into the formula, we get:
z = (4.4 - 4.7) / (2.4 / √64)
z = -0.3 / (2.4 / 8)
z = -0.3 / 0.3
z = -1
Now, we compare the calculated z-score (-1) to the critical value of the standard normal distribution corresponding to the desired level of significance (e.g., α = 0.05 for a 5% significance level).
If the calculated z-score is less than the critical value, we reject the null hypothesis in favor of the alternate hypothesis. Conversely, if the calculated z-score is greater than or equal to the critical value, we fail to reject the null hypothesis.
By looking up the critical value corresponding to α = 0.05 in the standard normal distribution (z-table or using statistical software), we find that the critical value is approximately -1.645.
Since the calculated z-score (-1) is greater than the critical value (-1.645), we fail to reject the null hypothesis. In other words, the sample data does not provide sufficient evidence to conclude that the mean distance from a home to the nearest fire department is less than 4.7 miles.
Therefore, based on the data and the hypothesis test, the null hypothesis (H0) that the mean distance from a home to the nearest fire department is at least 4.7 miles should be supported.