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

a. Identify the claim and state and


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. Step 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 price in Olathe is greater than in Rolla. So, let \(\mu_1\) be the mean price in Olathe and \(\mu_2\) be the mean price in Rolla. Then, the hypotheses are: \[ H_0: \mu_1 \leq \mu_2 \] \[ H_a: \mu_1 > \mu_2 \]
2. Step 2: Determine the significance level \(\alpha\), which is given as 0.05. This will be used to decide whether to reject the null hypothesis based on the test statistic.
3. Step 3: Since the population standard deviations are known, and the samples are independent, use the two-sample z-test for the difference of means. Calculate the test statistic using the formula: \[ z = \frac{(\bar{x}_1 - \bar{x}_2) - (\mu_1 - \mu_2)}{\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}} \] Here, \(\bar{x}_1 = 392,453\), \(\bar{x}_2 = 285,787\), \(\sigma_1 = 224,902\), \(\sigma_2 = 330,578\), \(n_1 = 39\), and \(n_2 = 38\). Under the null hypothesis, \(\mu_1 - \mu_2 = 0\).
4. Step 4: Find the critical z-value for a right-tailed test at \(\alpha = 0.05\). This value corresponds to the z-score where the area to the right is 0.05.
5. Step 5: Compare the calculated test statistic to the critical z-value. If the test statistic is greater than the critical value, reject the null hypothesis and conclude there is enough evidence to support the claim. Otherwise, do not reject the null hypothesis.

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 (α).

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 under the null hypothesis.

Significance Level and P-Value

The significance level (α) is the threshold for rejecting the null hypothesis, commonly set at 0.05. The p-value measures the probability of observing the sample data, or more extreme, assuming the null is true. If the p-value is less than α, the null hypothesis is rejected, supporting the alternative claim.