91Ó°ÊÓ

In 2018 Gallup reported that \(52 \%\) of Americans are dissatisfied with the quality of the environment in the United States. This was based on a \(95 \%\) confidence interval with a margin of error of 4 percentage points. Assume the conditions for constructing the confidence interval are met. a. Report and interpret the confidence interval for the population proportion that are dissatisfied with the quality of the environment in the United States in 2018 . b. If the sample size were larger and the sample proportion stayed the same, would the resulting interval be wider or narrower than the one obtained in part a? c. If the confidence level were \(90 \%\) rather than \(95 \%\) and the sample proportion stayed the same, would the interval be wider or narrower than the one obtained in part a? d. In 2018 the population of the United States was roughly 327 million. If the population had been half that size, would this have changed any of the confidence intervals constructed in this problem? In other words, if the conditions for constructing a confidence interval are met, does the population size have any effect on the width of the interval?

Step by step solution

01

Construction of the Confidence Interval

According to the exercise, 52% of Americans are dissatisfied, i.e., our sample proportion (\(p\)) is 0.52. We are also given the margin of error (ME) as 4% or 0.04, and confidence level as \(95 \%\). The confidence interval can be calculated as \(p\) \(\pm\) ME, so here it is \(0.52 \pm 0.04\). Therefore, the confidence interval is \(0.48\) to \(0.56\).

02

Effect of Larger Sample Size

The precision of a confidence interval is determined by the sample size. The larger the sample size, the more precise your estimates are, which leads to a narrower confidence interval, assuming the same level of confidence. So if the sample size were larger and the sample proportion stayed the same, the resulting interval would be narrower than the one computed in Step 1.

03

Effect of Lower Confidence Level

The width of a confidence interval is determined by the confidence level. A lower confidence level results in a narrower confidence interval, assuming the same sample size. So if the confidence level were \(90 \%\), the interval would be narrower than the one computed in Step 1.

04

Effect of Population Size on the Confidence Interval

The size of the population does not affect the width of the confidence interval assuming all other factors (sample proportion, sample size, and confidence level) remain the same. This is because confidence intervals depend on the variability within a sample, not the size of the population being studied. Thus, a decrease or increase in population size would not change the width of the confidence interval.

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