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.
Braking Distances To compare the dry braking distances from 60 to 0 miles per hour for two makes of automobiles, a safety engineer conducts braking tests for 16 compact SUVs and 11 midsize SUVs. The mean braking distance for the compact SUVs is 131.8 feet. Assume the population standard deviation is 5.5 feet. The mean braking distance for the midsize SUVs is 132.8 feet. Assume the population standard deviation is 6.7 feet. At α=0.10 , can the engineer support the claim that the mean braking distances are different for the two categories of SUVs? (Adapted from Consumer Reports)

Verified step by step guidance

1. Step 1: Identify the claim and state the null hypothesis (Ho) and alternative hypothesis (Ha). The claim is that the mean braking distances for compact SUVs and midsize SUVs are different. The null hypothesis (Ho) states that the mean braking distances are equal: Ho: μ1 = μ2. The alternative hypothesis (Ha) states that the mean braking distances are different: Ha: μ1 ≠ μ2.
2. Step 2: Determine the critical value(s) and rejection region(s). Since the test is two-tailed (due to the claim of 'difference'), use the significance level α = 0.10 to find the critical z-values. The rejection regions will be in the tails of the standard normal distribution, corresponding to the critical z-values.
3. Step 3: Calculate the standardized test statistic z. Use the formula for the z-test for two independent means: $z=\frac{(x_1-x_2)}{\sqrt{\frac{{{\sigma }_1}^2}{n_1}}}$, where x1 and x2 are the sample means, σ1 and σ2 are the population standard deviations, and n1 and n2 are the sample sizes.
4. Step 4: Compare the calculated z-value to the critical z-values. If the calculated z-value falls within the rejection region (outside the range defined by the critical z-values), reject the null hypothesis. Otherwise, fail to reject the null hypothesis.
5. Step 5: Interpret the decision in the context of the original claim. If the null hypothesis is rejected, conclude that there is sufficient evidence to support the claim that the mean braking distances are different for compact SUVs and midsize SUVs. If the null hypothesis is not rejected, conclude that there is insufficient evidence to support the claim.

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about population parameters based on sample data. It involves formulating two competing hypotheses: the null hypothesis (H0), which represents no effect or no difference, and the alternative hypothesis (Ha), which indicates the presence of an effect or difference. In this context, the engineer will test whether the mean braking distances of the two SUV categories differ significantly.

Critical Value and Rejection Region

The critical value is a threshold that determines the boundary for rejecting the null hypothesis in hypothesis testing. It is derived from the significance level (α), which indicates the probability of making a Type I error. The rejection region is the range of values for the test statistic that leads to the rejection of H0. In this case, with α=0.10, the engineer will identify the critical z-value to assess whether the observed test statistic falls within this region.

Standardized Test Statistic (z)

The standardized test statistic, often denoted as z, measures how many standard deviations an observed sample mean is from the hypothesized population mean under the null hypothesis. It is calculated using the means, standard deviations, and sample sizes of the groups being compared. In this scenario, the engineer will compute the z-value to evaluate the difference in mean braking distances between the compact and midsize SUVs, facilitating the decision to reject or fail to reject the null hypothesis.