Take this test as you would take a test in class.For each exercise, perform the steps below.
b.Determine whether the hypothesis test is left-tailed, right-tailed, or two-tailed, and whether to use a z-test or a t-test. Explain your reasoning.


A real estate agency says that the mean home sales price in Olathe, Kansas, is greater than in Rolla, Missouri. The mean home sales price for 39 homes in Olathe is \$392,453. Assume the population standard deviation is \$224,902. The mean home sales price for 38 homes in Rolla is \$285,787. Assume the population standard deviation is \$330,578. At α=0.05, is there enough evidence to support the agency’s claim? (Adapted from Realtor.com)

Verified step by step guidance

1. Identify the null hypothesis (H_0) and the alternative hypothesis (H_a). Since the agency claims that the mean home sales price in Olathe is greater than in Rolla, the hypotheses are: \(H_0: \mu_{Olathe} \leq \mu_{Rolla}\) \(H_a: \mu_{Olathe} > \mu_{Rolla}\) This indicates a right-tailed test because the claim is about one mean being greater than the other.
2. Determine whether to use a z-test or a t-test. Since the population standard deviations are known (\(\sigma_{Olathe} = 224,902\) and \(\sigma_{Rolla} = 330,578\)), and the sample sizes are reasonably large (39 and 38), a z-test is appropriate for comparing the two means.
3. Calculate the test statistic using the formula for the difference between two means with known population standard deviations: \(Z = \frac{(\bar{x}_1 - \bar{x}_2) - (\mu_1 - \mu_2)}{\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}}\) Here, \(\bar{x}_1\) and \(\bar{x}_2\) are the sample means, \(\sigma_1\) and \(\sigma_2\) are the population standard deviations, and \(n_1\) and \(n_2\) are the sample sizes. The hypothesized difference \((\mu_1 - \mu_2)\) under the null hypothesis is 0.
4. Determine the critical value for the right-tailed test at the significance level \(\alpha = 0.05\). This critical value corresponds to the z-score where the area to the right is 0.05.
5. Compare the calculated test statistic to the critical value. If the test statistic is greater than the critical value, reject the null hypothesis and conclude there is enough evidence to support the agency's claim. Otherwise, do not reject the null hypothesis.

Key Concepts

Hypothesis Testing and Tail Types

Hypothesis testing involves making a claim about a population parameter and using sample data to test it. The test can be left-tailed, right-tailed, or two-tailed depending on the claim: a right-tailed test checks if the mean is greater, left-tailed if less, and two-tailed if different. In this question, since the claim is that Olathe’s mean price is greater, a right-tailed test is appropriate.

Z-test vs. T-test

A z-test is used when the population standard deviation is known and the sample size is large (usually n > 30), while a t-test is used when the population standard deviation is unknown and/or the sample size is small. Here, population standard deviations are given and sample sizes are above 30, so a z-test is appropriate for comparing means.

Comparing Two Independent Sample Means

When comparing means from two independent samples, the test statistic measures the difference between sample means relative to the variability and sample sizes. The formula incorporates means, standard deviations, and sample sizes to determine if the observed difference is statistically significant at the chosen significance level (α = 0.05).