Question

Bank Breakup. In a nationwide survey, conducted by Pulse Opinion Research, LLC for Rasmussen Reports, a sample of American adults were asked whether they favor a plan to break up the $12$megabanks, which currently control about $69\%$of the banking industry; $50\%$of those sampled responded in the affirmative. According to the report, "the margin of sampling error is $\pm 3$percentage points with a $95\%$level of confidence." Find and interpret a $95\%$confidence interval for the percentage of all American adults who favor a plan to break up the $12$megabanks.

Short answer

The $95\%$ confidence interval for the proportion of all American adults who support breaking up the $12$ megabanks is $0.47$ to $0.53$.

Step-by-step solution

Given Information

A sample of American adults were asked whether they favour a plan to break up the $12$ megabanks, which currently control about $69\%$ of the banking industry; $50\%$ of those sampled responded in the affirmative. According to the report," the margin of sampling error is $+/-3$ percentage points with a $95\%$ level of confidence."

Explanation

When $x$and $n-x$are both $5$the one-proportion z-interval technique is appropriate if the number of proportions is larger than one.

It has a $95\%$this is referred to as a confidence interval. $\alpha =0.05$

It finds that $z_{\alpha /2}=z_{0.05/2}$

$=1.96$

The range of confidence for $p$is form

$p^{\text{'}}=0.50$

Margin error $E=3\%$.

The $95\%$of confidence interval is

$p^{\text{'}}\pm E=0.50\pm 0.03$

$=0.50\pm 0.03$

$(0.47,0.53)$

Thus, the localid="1651311809459" $95\%$Confidence interval for the percentage of all adults in the United States who support a proposal to break up the $12$megabanks is $(0.47,0.53)$.