Question

Suppose that county planners are interested in learning about the proportion of county residents who would pay a fee for a curbside recycling service if the county were to offer this service. Two different people independently selected random samples of county residents and used their sample data to construct the following confidence intervals for the proportion who would pay for curbside recycling: Interval 1:(0.68,0.74) Interval 2:(0.68,0.72) 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 are associated with 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

The centers of confidence intervals may differ due to the independent random selection of samples. Interval 2, being narrower, presents a more precise population proportion. With a \(95\% \) confidence level, Interval 1 was likely derived from a smaller sample size, as wider intervals usually indicate more variability due to a smaller sample. If the sample size was identical for both intervals, Interval 2 would have a higher confidence level due to its reduced width.

Step-by-step solution

Explain differing center points

The centers of two confidence intervals can differ because samples were taken independently. Even though each is a random sample of the general population, differences in the individuals selected for each sample can lead to variability in the sample proportions and, henceforth, the centres of the confidence intervals.

Comparing precision

Interval 2 (0.68, 0.72) can be said to provide more precise information about the population proportion. The reasoning for this is that it is narrower than Interval 1 (0.68, 0.74), meaning there is less uncertainty about the true population proportion.

Determining sample size

As the confidence level for both intervals is \(95\% \), the interval with the wider range (Interval 1) was founded on a smaller sample size. The range of a confidence interval is inversely related to the sample size; as the sample size decreases, the variability increases, leading to a broader confidence interval.

Comparing Confidence Levels

If both intervals were based on the same sample size, Interval 2 (0.68, 0.72) would have a higher confidence level. The reason being that with a narrower interval, we are more certain about the population proportion falling within this particular range.