Question

the headline reporting the results of a poll conducted by the gallup organization stated, \majority of americans at personal best in the morning.\ the results indicated that a survey of 1100 adults resulted in 55% stating they were at their personal best in the morning. the polls results were reported with a margin of error of 3%. explain why the gallup organizations headline is accurate.

choose the correct answer below.

a. more than 30 people were surveyed.

b. all the values within the margin of error are greater than 50%.

c. all the values within the margin of error are less than 50%.

d. more than 50% of the respondents answered in the affirmative.

Explanation:

The margin of error is \(3\%\). The sample proportion is \(55\%\). The confidence interval is calculated as \(\hat{p}\pm E\), where \(\hat{p} = 0.55\) and \(E=0.03\). The lower bound of the confidence interval is \(0.55 - 0.03=0.52\) (or \(52\%\)) and the upper bound is \(0.55 + 0.03 = 0.58\) (or \(58\%\)). Since all values within the confidence interval (\(52\%\) to \(58\%\)) are greater than \(50\%\), it can be said that a majority (more than \(50\%\)) of Americans are at their personal best in the morning.

Answer:

B. All the values within the margin of error are greater than \(50\%\)