Each year the US Environmental Protection Agency (EPA) releases fuel economy data on cars manufactured in that year. Below are summary statistics on fuel efficiency (in miles/gallon) from random samples of small cars with manual and automatic transmissions. Which of the following statements is true if we were to decide whether or not these data provide strong evidence of a difference between the average fuel efficiency of cars with manual and automatic transmissions in terms of their average city mileage? Assume that conditions for inference are satisfied.

Automatic Manual
Mean 32.4 28.8
Standard Deviation 2.12 5.75
n 27 26

Question

Each year the US Environmental Protection Agency (EPA) releases fuel economy data on cars manufactured in that year. Below are summary statistics on fuel efficiency (in miles/gallon) from random samples of small cars with manual and automatic transmissions. Which of the following statements is true if we were to decide whether or not these data provide strong evidence of a difference between the average fuel efficiency of cars with manual and automatic transmissions in terms of their average city mileage? Assume that conditions for inference are satisfied.

Automatic	Manual
Mean	32.4	28.8
Standard Deviation	2.12	5.75
n	27	26

BlackTom AI · Accepted Answer

Let's carefully examine each statement about the two-sample inference scenario described.

Option 1: 'The degrees of freedom for this hypothesis test is 25.'
- In a two-sample t-test, the degrees of freedom depend on which version of the test we use. If we pool the variances (assuming equal variances), the standard two-sample t-test uses df = n1 + n2 − 2, which here would be 27 + 26 − 2 = 51. If we do not assume equal variances (the Welch t-test), the degrees of freedom are computed by a more complicated formula that typically yields a value between the smaller of (n1 − 1) and (n2 − 1) and something larger, often not an integer, and certainly not a fixed 25 in this setup. Specifically, min(n1 − 1, n2 − 1) = min(26, 25) = 25 provides only a rough lower bound in some approximations, but the exact Welch df would normally be different and is not simply 25 here. Thus, asserting df = 25 as a universal truth for this test is not accurate given standard procedures. This option is likely false under standard inference methods.

Option 2: 'When calculating the standard error, we first pool the standard deviations from both samples because we assume no differences in variabilities between the two samples.'
- Pooling the standard deviations is the hallmark of the equal-variances (pooled) two-sample t-test. This approach assumes that the population variances are the same, which is a strong assumption. Looking at the data provided, the sample standard deviations are 2.12 (Automatic) and 5.75 (Manual), which are quite different relative to their means. Such a large discrepancy in sample SDs strongly suggests unequal variances, so pooling would misrepresent the true variability. Therefore, this statement is incorrect because it claims pooling is appropriate when the evidence suggests otherwise.

Option 3: 'A paired t-test is the appropriate test to use.'
- A paired t-test applies when observations are matched or the data come in naturally paired form (e.g., before-and-after measurements on the same subject, or matched pairs). Here, the data come from two independent samples: cars with automatic transmissions and cars with manual transmissions. There is no pairing between individual automatic-car observations and manual-car observations. Consequently, a paired t-test is not appropriate for these data. This statement is incorrect due to the independence of samples.

Option 4: 'We can assume equal variances because the sample sizes are similar for both samples.'
- Even though sample sizes are close (n1 = 27, n2 = 26), equal-variance is not warranted merely by similar sample sizes. The key evidence here is the substantial difference between the sample standard deviations (2.12 vs 5.75). If the population variances were truly equal, we would expect the sample SDs to be more similar, especially with reasonably large samples. The observed disparity indicates that the equal-variances assumption is questionable, and relying on it would likely distort inference. Therefore, this statement is incorrect as a general rule when variances appear unequal.

In summary, each of the four statements contains a flaw when confronted with the data and standard inference practices for two independent samples. The first statement attempts to pin a specific df value that does not align with the common methods given the unequal variances and the Welch versus pooled-variance considerations; the second emphasizes pooling despite evident variance differences; the third incorrectly treats the situation as paired data; and the fourth wrongly asserts equal variances based on similar sample sizes alone. The overall reasoning highlights why the standard approach would not support these particular claims under normal two-sample inference assumptions.

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Similar Questions

A two-sided independent samples t-test with n1=33 and n2=38 yielded a test statistic of 2.03. Assuming unequal variances, which of the following is the correct p-value for this hypothesis test? Be sure to find the p-value using the t-distribution probability table.

When should we used pooled variance for a test of two sample means?

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