"Take this test as you would take a test in class.For each exercise, perform the steps below.
f. Interpret the decision in the context of the original claim.
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 and alternative hypotheses based on the claim. Since the agency claims that the mean home sales price in Olathe is greater than in Rolla, set up the hypotheses as: \(H_0: \mu_{Olathe} \leq \mu_{Rolla}\) \(H_a: \mu_{Olathe} > \mu_{Rolla}\)
2. Determine the significance level \(\alpha = 0.05\) and note the sample sizes, means, and population standard deviations: \(n_1 = 39\), \(\bar{x}_1 = 392,453\), \(\sigma_1 = 224,902\) \(n_2 = 38\), \(\bar{x}_2 = 285,787\), \(\sigma_2 = 330,578\)
3. Calculate the test statistic for the difference between two means when population standard deviations are known, using the formula: \(Z = \frac{(\bar{x}_1 - \bar{x}_2) - 0}{\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}}\) Note that the hypothesized difference under \(H_0\) is 0.
4. Find the critical value for a right-tailed test at \(\alpha = 0.05\) from the standard normal distribution (Z-distribution). This critical value will be used to compare with the calculated test statistic.
5. Make a decision by comparing the test statistic to the critical value: - If \(Z\) is greater than the critical value, reject \(H_0\) and conclude there is enough evidence to support the claim. - Otherwise, fail to reject \(H_0\) and conclude there is not enough evidence to support the claim. Finally, interpret this decision in the context of the problem, explaining what it means about the mean home sales prices in Olathe and Rolla.

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to decide whether there is enough evidence to support a specific claim about a population parameter. It involves formulating a null hypothesis (no effect or difference) and an alternative hypothesis (the claim), then using sample data to determine if the null can be rejected at a given significance level.

Two-Sample Z-Test for Means

A two-sample z-test compares the means of two independent populations when population standard deviations are known. It calculates a test statistic based on sample means, standard deviations, and sizes to assess if the difference between means is statistically significant, helping to evaluate claims about population means.

Significance Level and Decision Rule

The significance level (α) is the threshold probability for rejecting the null hypothesis, commonly set at 0.05. It defines the risk of a Type I error (false positive). The decision rule compares the test statistic to critical values or p-values to determine if the evidence is strong enough to support the alternative hypothesis.