Testing the Difference Between Two Means In Exercises 15–24, (a) identify the claim and state Ho and Ha, (b) find the critical value(s) and identify the rejection region(s), (c) find the standardized test statistic z, (d) decide whether to reject or fail to reject the null hypothesis, and (e) interpret the decision in the context of the original claim. Assume the samples are random and independent, and the populations are normally distributed.
Home Prices A real estate agency says that the mean home sales price in Casper, Wyoming, is the same as in Cheyenne, Wyoming. The mean home sales price for 35 homes in Casper is \$349,237. Assume the population standard deviation is \$158,005. The mean home sales price for 41 homes in Cheyenne is \$435,244. Assume the population standard deviation is \$154,716. At α=0.01, is there enough evidence to reject the agency’s claim? (Adapted from Realtor.com)

Verified step by step guidance

1. Identify the claim and state the null hypothesis (H\_0) and the alternative hypothesis (H\_a). The claim is that the mean home sales prices in Casper and Cheyenne are the same. So, H\_0: \(\mu\)_1 = \(\mu\)_2 (means are equal) and H\_a: \(\mu\)_1 \(\neq\) \(\mu\)_2 (means are different), where \(\mu\)_1 is the mean price in Casper and \(\mu\)_2 is the mean price in Cheyenne.
2. Find the critical value(s) for a two-tailed test at the significance level \(\alpha\) = 0.01. Since the test is two-tailed, find the z-values that correspond to the upper and lower 0.5% tails of the standard normal distribution. These z-values define the rejection regions.
3. Calculate the standardized test statistic z 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. Since H\_0 assumes \(\mu\)_1 - \(\mu\)_2 = 0, substitute that in the numerator.
4. Compare the calculated z-value to the critical values found in step 2. If z falls into the rejection region (less than the lower critical value or greater than the upper critical value), reject the null hypothesis. Otherwise, fail to reject the null hypothesis.
5. Interpret the decision in the context of the original claim. If you rejected H\_0, conclude that there is sufficient evidence at the 0.01 significance level to say the mean home prices in Casper and Cheyenne are different. If you failed to reject H\_0, conclude that there is not enough evidence to say the means differ, supporting the agency's claim.

Key Concepts

Hypothesis Testing for Two Means

This involves comparing the means of two independent samples to determine if there is a statistically significant difference between their population means. The null hypothesis (Ho) typically states that the means are equal, while the alternative hypothesis (Ha) states they are different. This framework guides the testing process and decision-making.

Z-Test for Two Independent Samples with Known Population Standard Deviations

When population standard deviations are known and samples are independent, the z-test is used to calculate the standardized test statistic. This involves computing the difference between sample means, divided by the standard error derived from the population standard deviations and sample sizes. The z-test helps determine how extreme the observed difference is under the null hypothesis.

Critical Value and Rejection Region

The critical value is a threshold from the standard normal distribution corresponding to the chosen significance level (α). It defines the rejection region(s) where the null hypothesis is rejected if the test statistic falls within these regions. For a two-tailed test at α=0.01, critical values mark the extreme 0.5% tails on both ends of the distribution.