Question

Samples of two different types of automobiles were selected, and the actual speed for each car was determined when the speedometer registered \(50 \mathrm{mph}\). The resulting \(95 \%\) confidence intervals for true average actual speed were \((51.3,52.7)\) and \((49.4,50.6)\). Assuming that the two sample standard deviations are identical, which confidence interval is based on the larger sample size? Explain your reasoning.

Short answer

The confidence interval \((49.4,50.6)\) for the second type of automobile is based on a larger sample size. This conclusion is reached by comparing the widths of the given confidence intervals.

Step-by-step solution

Understand the Confidence Intervals

The confidence intervals given for the two types of automobiles are \( (51.3,52.7) \) and \( (49.4,50.6) \). The width of these intervals gives us an idea of the precision of our estimate. The narrower the interval, the more precise our estimate is.

Compare the Width of the Confidence Intervals

To know which confidence interval is based on a larger sample size, we compare their width. The width of a confidence interval for the mean is largely determined by the standard deviation and the sample size. Because we are assuming the standard deviations to be identical, the interval with the smaller width should have a larger sample size. The width of the first interval is \(52.7 - 51.3 = 1.4\) and that of the second interval is \(50.6 - 49.4 = 1.2\)

Decide which sample size is larger

Since the width of the second interval (1.2) is less than the width of the first interval (1.4), this suggests that the second confidence interval is based on a larger sample size. This is because the greater the size of the sample, the lower the variability around the mean, resulting in a narrower confidence interval.