Question

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)

Accepted Answer

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$$: \mu_{traditional} \u003e \mu_{box}$$ 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$. Calculate the standardized test statistic z$ 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. 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$. Otherwise, fail to reject H_0$. 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.

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Hypothesis Testing for Two Means

Critical Value and Rejection Region

Standardized Test Statistic (z-score)