Question

1. two researchers, researcher a and researcher b, independently survey random samples of 1,000 and 500 college students, respectively, to estimate the proportion of all college students who prefer online courses to in - person courses. both researchers use the data they obtain to construct a 99% confidence interval. these intervals are given in the table below. assuming the researchers did not make any calculation errors when constructing their confidence intervals, which one of the following statements is correct?

a. researcher as interval is more accurate than researcher bs interval since researcher a surveyed a larger sample.
b. the smaller sample size is the one that resulted in a narrower confidence interval width.
c. we have enough information here to verify that the intervals were computed correctly.
d. the widths of the two intervals are different because of differences in population size.
e. changing the level of confidence to 95% will result in wider intervals.

Explanation:

Step1: Calculate the width of Researcher A's interval

The width of a confidence interval is calculated as \(upper - lower\). For Researcher A, the width is \(0.562 - 0.480=\ 0.082\)

Step2: Calculate the width of Researcher B's interval

For Researcher B, the width is \(0.563 - 0.447 = 0.116\)

Step3: Analyze each option

• Option A:

The formula for the margin of error \(E = z_{\alpha/2}\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}\) (where \(n\) is the sample size). A larger sample size \(n\) leads to a smaller margin of error (assuming the same confidence level and similar \(\hat{p}\)). Researcher A has a larger sample size (\(n_A=1000\), \(n_B = 500\)). A smaller margin of error means a more accurate estimate (narrower interval). Researcher A's interval width (\(0.082\)) is narrower than Researcher B's (\(0.116\)).

• Option B:

As shown in Step1 and Step2, the larger sample size (Researcher A with \(n = 1000\)) resulted in a narrower confidence - interval width.

• Option C:

To verify if the intervals were computed correctly, we would need to know the sample proportion \(\hat{p}\) (since the formula for the confidence interval for a proportion is \(\hat{p}\pm z_{\alpha/2}\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}\)). We do not have information about \(\hat{p}\) for either sample.

• Option D:

The formula for the margin of error for a proportion \(E=z_{\alpha/2}\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}\) does not depend on the population size (when the sample is a small fraction of the population, which is usually the case in such surveys). The difference in widths is due to the difference in sample sizes.

• Option E:

A lower confidence level (e.g., changing from \(99\%\) to \(95\%\)) will result in a smaller \(z\) - value (\(z_{0.005}=2.576\) for \(99\%\) confidence and \(z_{0.025}=1.96\) for \(95\%\) confidence). A smaller \(z\) - value will result in a narrower confidence interval.

Answer:

A. Researcher A’s interval is more accurate than Researcher B’s interval since Researcher A surveyed a larger sample.