Question

Justice in the Courts? In an issue of Parade Magazine, the editors reported on a national survey on law and order. One question asked of the $2512$ U.S. adults who took part was whether the believed that juries "almost always" convict the guilty and free the innocent. Only $578$ said that they did. At the $5\%$ significance level do the data provide sufficient evidence to conclude that less than one in four Americans believe that juries "almost always" convict the guilty and free the innocent?

Short answer

The data shows that less than one-quarter of all individuals in the United States believe juries "usually invariably" convict the guilty and exonerate the innocent.

Step-by-step solution

Given Information

Sample size is $n=2512$

Number of success is $x=578$.

Sample proportion would be $\hat{p}=\frac{x}{n}$

$=\frac{578}{2512}$

$=0.1107$

Explanation

A significance level of $\alpha$,

To find $z_{\alpha /2}$using table II.

Significance level is $5\%$.

i.e. $\alpha =0.05$.

Using table II,

$z_{\alpha /2}=z_{0.05/2}$

$=z_{0.025}$

$=1.96$

Explanation

The confidence interval for $p$is:

$\hat{p}\pm z_{\alpha /2}·\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$

$\Rightarrow \mathrm{CI}=0.1107\pm 1.96\sqrt{\frac{0.1107(1-0.1107)}{2512}}$

$\Rightarrow \mathrm{CI}=0.1107\pm 0.0127$

$\Rightarrow \mathrm{CI}=0.0980\text{to}0.1234$

The highest limit of confidence is less than $0.25$at a $5\%$significance level (one in four).