Take this test as you would take a test in class.For each exercise, perform the steps below.

e. Decide whether to reject or fail to reject the null hypothesis.


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, set up the hypotheses as: \(H_0: \mu_{Olathe} \leq \mu_{Rolla}\) \(H_a: \mu_{Olathe} > \mu_{Rolla}\)
2. Determine the significance level, which is given as \(\alpha = 0.05\). This will be used to decide the rejection region for the test.
3. Calculate the test statistic for the difference between two means when population standard deviations are known. Use the formula: \(Z = \frac{(\bar{x}_1 - \bar{x}_2) - (\mu_1 - \mu_2)}{\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}}\) where \(\bar{x}_1\) 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. Since under the null hypothesis \(\mu_1 - \mu_2 = 0\), simplify accordingly.
4. Find the critical value from the standard normal distribution corresponding to \(\alpha = 0.05\) for a one-tailed test (right tail). This critical value will be the cutoff point to decide whether to reject the null hypothesis.
5. Compare the calculated test statistic to the critical value: - If the test statistic is greater than the critical value, reject the null hypothesis. - Otherwise, fail to reject the null hypothesis. This decision will tell you if there is enough evidence at the 5% significance level to support the agency's claim.

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 whether to reject or fail to reject the null hypothesis based on a 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 z-score to measure the difference between sample means relative to the variability, helping determine if the observed difference is statistically significant at a given significance level.

Significance Level and Decision Rule

The significance level (α) is the threshold probability for rejecting the null hypothesis, commonly set at 0.05. If the test statistic falls into the critical region defined by α, we reject the null hypothesis; otherwise, we fail to reject it. This controls the risk of a Type I error, which is falsely claiming a difference exists.