Question

A poll conducted by Gallup in December $2013$asked a sample of American adults whether they approved of the way President Obama was doing his job; $42\%$said yes, with a margin of error of plus or minus $3$percentage points. During that same time period, Quinnipiac University asked the same question of a sample of American adults; $38\%$said yes, with a margin of error of plus or minus$2$ percentage points. Can the conclusions of both polls be correct? Explain your answer.

Short answer

It's possible that the results of both polls are correct.

Step-by-step solution

Given Information

According to a Gallup poll, $42\%$of American adults approve of the way President Obama is handling his job (with a margin of error of plus or minus $3\%$points).

According to a second poll taken by Quinnipiac University, $38\%$ of American adults approved of President Obama's job performance (with a margin of error of plus or minus $2\%$ points).

Explanation

Assume that both surveys utilized a confidence level of $95\%$. As a result, both polls give the percentage/proportion of American adults who believe President Obama is performing his job correctly at a $95\%$confidence level.

According to the information provided,

The Gallup poll's confidence interval is $39\%\text{to}45\%(42\pm 3)$.

Quinnipiac University's confidence interval is $36\%\text{to}40\%(38\pm 2)$.

It should be emphasized that both polls give common believable results ranging from$39\%$to$40\%$.