Question

Suppose that a campus bookstore manager wants to know the proportion of students at the college who purchase some or all of their textbooks online. Two different people independently selected random samples of students at the college and used their sample data to construct the following confidence intervals for the population proportion: Interval 1:(0.54,0.57) Interval 2:(0.46,0.62) a. Explain how it is possible that the two confidence intervals are not centered in the same place. b. Which of the two intervals conveys more precise information about the value of the population proportion? c. If both confidence intervals have a \(95 \%\) confidence level, which confidence interval was based on the smaller sample size? How can you tell? d. If both confidence intervals were based on the same sample size, which interval has the higher confidence level? How can you tell?

Short answer

a) The intervals have different centers because they're based on different samples with different sample proportions. b) The Interval 1 conveys more precise information because it is narrower, i.e., width of 0.03 compared to Interval 2's width of 0.16. c) Between the two intervals, Interval 2 is wider and, therefore, was based on the smaller sample size. d) If both intervals were based on same sample size, Interval 2 has a higher confidence level since it is wider.

Step-by-step solution

Comparative Analysis of Confidence Intervals' Centers

Understand that the center of a confidence interval is the point estimate, which is the sample proportion. In this case, the point estimates for Interval 1 and Interval 2 are different, which simply means the samples collected by the two people had different proportions of students who purchased textbooks online.

Precision of the Sample Proportions

Recognize that a tighter or narrower interval would give more precise estimates; this means the interval with smaller width is more precise. Calculate the widths by subtracting each interval's lower limit from its upper limit. The smaller interval width indicates more precise information about the population proportion.

Sample Size and Confidence Interval

Notice that for a fixed confidence level, the width of the confidence interval decreases as the sample size increases. Therefore, the interval with smaller width was based on the larger sample size.

Confidence Level and Confidence Interval

Understand that the wider an interval for a certain sample size, the higher the confidence level. So if both intervals were based on the same sample size, the wider interval would have a higher confidence level.