Question

The headline reporting the results of a poll​ stated, "Majority of Adults at Personal Best in the​ Morning." The results indicated that a survey of 1000 adults resulted in 57​% stating they were at their personal best in the morning. The​ poll's results were reported with a margin of error of 2​%. Explain why the​ poll's 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. More than​ 50% of the respondents answered in the affirmative.

Answer 1

Based on the given data, a 100(1-)% CI for proportion is given by,

where, is called the Margin of Error.

Given:

The confidence interval is given by,

We find that the true population proportion would lie in the inetrval , which is greater than 50%.  

A. It is the number of favorable cases that determines the proportion of Adults who state that they are at their Personal Best in the​ Morning and not the Total Sample size (n). Hence, "More than 30 people were surveyed" cannot be an evidence for the accuracy of the poll's estimate.

B. Here, since we are dealing count data and not the values of individual observations as in the claim "All the values within the margin of error are greater than​ 50%". Since, this is a bivariate, the observation value would simply be a o (If Yes) or 0 (If No). Here, all values cannot be greater than 50%, unless every person agrees to "Personal Best in the​ Morning", which is not the case here.

C. It is known that the Sample proportion is an unbiased estimate of Population Proportion. Given that = 0.57 and the confidence interval for the true proportion , we find that More than​ 50% of the respondents answered in the affirmative.Hence, this would be the correct reason to believe that the poll's headline is accurate.

Ans. Option C.