Question

Samples of two different models of cars were selected, and the actual speed for each car was determined when the speedometer registered \(50 \mathrm{mph}\). The resulting \(95 \%\) confidence intervals for mean actual speed were (51.3,52.7) for model 1 and (49.4,50.6) for model 2 . Assuming that the two sample standard deviations were equal, which confidence interval is based on the larger sample size? Explain your reasoning.

Short answer

The confidence interval that is based on the larger sample size is that of model 2 because it has a smaller confidence interval.

Step-by-step solution

Computing the Interval Length for Model 1

Calculate the length of the confidence interval for the mean speed of model 1. It will be found by subtracting the lower limit of the interval from the upper limit. In this case, calculate \(52.7 - 51.3 = 1.4\).

Computing the Interval Length for Model 2

Calculate the length of the confidence interval for the mean speed of model 2. Still, it's found by subtracting the lower limit from the upper limit. Now, calculate \(50.6 - 49.4 = 1.2\).

Comparison of Sample Sizes

Now, by comparing the lengths of these two intervals, it is evident that the confidence interval of model 2 is shorter than that of model 1. Therefore, it is concluded that the sample size for model 2 is larger than the sample size for model 1. The reason is that for a fixed level of confidence, the precision of the estimate (measured by the width of the confidence interval) increases with the sample size. This means a larger sample size will produce a narrower, more precise confidence interval.