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

c.Find the critical value(s) and identify the rejection region(s).




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 type of test and the hypotheses. Since the agency claims that the mean home sales price in Olathe is greater than in Rolla, this is a right-tailed test. The null hypothesis is \(H_0: \mu_{Olathe} \leq \mu_{Rolla}\) and the alternative hypothesis is \(H_a: \mu_{Olathe} > \mu_{Rolla}\).
2. Determine the significance level \(\alpha = 0.05\) and the type of test statistic to use. Because the population standard deviations are known, use the Z-test for the difference between two means.
3. Find the critical value for a right-tailed Z-test at \(\alpha = 0.05\). This critical value corresponds to the Z-score where the area to the right is 0.05.
4. Define the rejection region based on the critical value. For a right-tailed test, the rejection region is all Z-scores greater than the critical value.
5. Summarize: If the calculated test statistic exceeds the critical value, reject the null hypothesis; otherwise, do not reject it.

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.

Critical Value and Rejection Region

The critical value is a threshold that defines the boundary of the rejection region in hypothesis testing. It depends on the significance level (α) and the test type (one-tailed or two-tailed). If the test statistic falls into the rejection region beyond the critical value, the null hypothesis is rejected.

Two-Sample Z-Test for Means with Known Population Standard Deviations

This test compares the means of two independent samples when population standard deviations are known. It calculates a Z statistic based on sample means, population standard deviations, and sample sizes to assess if the difference between means is statistically significant.