Question

A July 2011 Roper Poll showed that 59% of its randomly selected U.S. adults would support banning smoking in all public places. (This was up from 44% in July 2010.) Suppose the margin of error was 4 percentage points with a 95% confidence interval. Assume the conditions hold for all parts.

a. Report the confidence interval for the population percentage that supported banning smoking in all public places in 2011, using a carefully worded sentence.
b. If the sample size were smaller and the sample proportion stayed the same, would the interval be wider or narrower than the one obtained in part a?
c. If the confidence level were 90% and nothing else changed, would the interval be wider or narrower than the one obtained in part a?
d. The total population of adults in the United States in 2011 was about 238 million. Suppose the population had been about one-quarter of that.
Would that have changed any of your answers?

Answer 1

The 95% confidence interval for the 2011 poll is between 55% and 63%. A smaller sample size would lead to a wider interval, while a lower confidence level, such as 90%, would result in a narrower interval. The total population size does not significantly impact the confidence interval.

Step-by-step explanation:

a. The confidence interval for the population percentage that supported banning smoking in all public places in 2011, considering the margin of error is ±4 percentage points and the poll result was 59%, would be between 55% (59% - 4%) and 63% (59% + 4%). So, we are 95% confident that the true proportion of U.S. adults who supported banning smoking in all public places in July 2011 is between 55% and 63%.

b. If the sample size were smaller with the sample proportion remaining the same, the resulting confidence interval would be wider. Smaller sample sizes lead to larger margins of error since the estimate would be less precise.

c. With a 90% confidence level, the interval would be narrower than one calculated at a 95% confidence level. Lower confidence levels reduce the margin of error.

d. The total population size does not usually affect the width of the confidence interval once the population is large enough. This concept is known as the 'finite population correction', which is negligible in large populations such as 238 million or one-quarter of that size. Hence, none of the answers would have changed if the population had been about one-quarter of 238 million.