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.
Bed-in-a-Box To compare customer satisfaction with mattresses that are delivered compressed in a box and traditional mattresses, a researcher randomly selects 30 ratings of mattresses in boxes and 30 ratings of traditional mattresses. The mean rating of mattresses in boxes is 68.7 out of 100. Assume the population standard deviation is 6.6. The mean rating of traditional mattresses is 70.9 out of 100. Assume the population standard deviation is 5.6. At α=0.01, can the researcher support the claim that the mean rating of traditional mattresses is greater than the mean rating of mattresses in a box? (Adapted from Consumer Reports)

Verified step by step guidance

1. Identify the claim and state the null hypothesis (H\_0) and alternative hypothesis (H\_a). Since the claim is that the mean rating of traditional mattresses is greater than that of mattresses in a box, set up: \(H_0: \mu_{traditional} \leq \mu_{box}\) \(H_a: \mu_{traditional} > \mu_{box}\)
2. Determine the significance level \(\alpha = 0.01\) and find the critical value for a right-tailed z-test. Use the standard normal distribution table to find the z-value such that the area to the right is 0.01. This z-value will define the rejection region where if the test statistic exceeds it, you reject \(H_0\).
3. Calculate the standardized test statistic \(z\) using the formula for the difference between two means with known population standard deviations: \(z = \frac{(\bar{x}_{traditional} - \bar{x}_{box}) - 0}{\sqrt{\frac{\sigma_{traditional}^2}{n_{traditional}} + \frac{\sigma_{box}^2}{n_{box}}}}\) where \(\bar{x}\) are the sample means, \(\sigma\) are the population standard deviations, and \(n\) are the sample sizes.
4. Compare the calculated \(z\) value to the critical value found in step 2. If \(z\) is greater than the critical value, reject the null hypothesis \(H_0\). Otherwise, fail to reject \(H_0\).
5. Interpret the decision in the context of the problem: If you rejected \(H_0\), conclude that there is sufficient evidence at the 0.01 significance level to support the claim that the mean rating of traditional mattresses is greater than that of mattresses in a box. If you failed to reject \(H_0\), conclude that there is not enough evidence to support the claim.

Key Concepts

Hypothesis Testing for Two Means

This involves comparing the means of two independent populations to determine if there is a statistically significant difference. The null hypothesis (Ho) typically states that the means are equal, while the alternative hypothesis (Ha) reflects the claim, such as one mean being greater than the other. This framework guides the testing process.

Critical Value and Rejection Region

The critical value is a threshold determined by the significance level (α) and the test type (one-tailed or two-tailed). It defines the rejection region, where if the test statistic falls within, the null hypothesis is rejected. For α=0.01 and a one-tailed test, the critical z-value marks the boundary for strong evidence against Ho.

Standardized Test Statistic (z-score)

The z-score measures how many standard errors the observed difference between sample means is from the hypothesized difference (usually zero). It is calculated using sample means, population standard deviations, and sample sizes. Comparing this z to the critical value helps decide whether to reject Ho.