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.
Repair Costs: Washing Machines You want to buy a washing machine, and a salesperson tells you that the mean repair costs for Model A and Model B are equal. You research the repair costs. The mean repair cost of 24 Model A washing machines is \$208. Assume the population standard deviation is \$18. The mean repair cost of 26 Model B washing machines is \$221. Assume the population standard deviation is \$22. At α=0.01, can you reject the salesperson’s claim?

Verified step by step guidance

1. Identify the claim and state the null hypothesis (H\_0) and alternative hypothesis (H\_a). The claim is that the mean repair costs for Model A and Model B are equal. So, H\_0: \(\mu\)\_A = \(\mu\)\_B and H\_a: \(\mu\)\_A \(\neq\) \(\mu\)\_B (two-tailed test).
2. Find the critical value(s) for a two-tailed test at significance level \(\alpha\) = 0.01. Use the standard normal distribution (z-distribution) because population standard deviations are known. Determine the z-values that correspond to the upper and lower 0.5% tails (since 0.01 total significance level is split between two tails).
3. Calculate the standardized test statistic z using the formula: \( z = \frac{(\bar{x}_A - \bar{x}_B) - (\mu_A - \mu_B)}{\sqrt{\frac{\sigma_A^2}{n_A} + \frac{\sigma_B^2}{n_B}}} \) Here, \(\bar{x}\)_A and \(\bar{x}\)_B are the sample means, \(\sigma\)_A and \(\sigma\)_B are the population standard deviations, and n_A and n_B are the sample sizes. Since H\_0 assumes \(\mu\)_A - \(\mu\)_B = 0, substitute that in the numerator.
4. Compare the calculated z statistic 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 repair costs for Model A and Model B are different. If you failed to reject H\_0, conclude that there is not sufficient evidence to dispute the salesperson's claim that the mean repair costs are equal.

Key Concepts

Hypothesis Testing for Two Means

Hypothesis testing for two means involves comparing the average values from two independent samples 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) states they are not. This framework helps decide if observed differences are due to chance or reflect true population differences.

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

When population standard deviations are known and samples are independent, a z-test can be used to compare two means. The test statistic z measures how many standard errors the sample mean difference is from zero. It is calculated using the formula involving sample means, population standard deviations, and sample sizes, assuming normality of populations.

Significance Level, Critical Values, and Rejection Regions

The significance level (α) defines the probability of rejecting the null hypothesis when it is true (Type I error). Critical values are z-scores that mark the boundaries of the rejection region(s) in the sampling distribution. If the test statistic falls into these regions, the null hypothesis is rejected, indicating sufficient evidence against the claim.