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Expert-verified Found in: Page 640 ### The Practice of Statistics for AP Examination

Book edition 6th
Author(s) Daren Starnes, Josh Tabor
Pages 837 pages
ISBN 9781319113339 # Where’s Egypt? In a Pew Research poll, $287$ out of $522$randomly selected U.S. men were able to identify Egypt when it was highlighted on a map of the Middle East. When $520$ randomly selected U.S. women were asked, $233$ were able to do so.a. Construct and interpret a $95%$ confidence interval for the difference in the trueproportion of U.S. men and U.S. women who can identify Egypt on a map.b. Based on your interval, is there convincing evidence of a difference in the trueproportions of U.S. men and women who can identify Egypt on a map? Justify youranswer.

a. Confidence Interval is $\left(-0.0191,0.1017\right)$.

b. Yes, there is evidence of difference in true proportion of US men and women who can identify Egypt on map or not.

See the step by step solution

## Step 1: Given Information

It is given that ${x}_{1}=287$

${x}_{2}=233$

${n}_{1}=522$

${n}_{2}=520$

$c=95%=0.95$

## Step 2: Calculating Confidence Interval

The three conditions are:

Random: Samples are independent random samples.

Independent: $522$ US men are less than $10%$ of population. Same is true for women.

Normal: Success in two samples are $287,233$ and failures are $235,287$. All are greater than ten.

All conditions are satisfied.

Sample Proportion: ${\stackrel{^}{p}}_{1}=\frac{{x}_{1}}{{n}_{1}}=\frac{287}{522}=0.5498$

${\stackrel{^}{p}}_{2}=\frac{{x}_{2}}{{n}_{2}}=\frac{233}{520}=0.4481$

Confidence Interval:

$\left({\stackrel{^}{p}}_{1}-{\stackrel{^}{p}}_{2}\right)-{z}_{\alpha /2}×\sqrt{\frac{{\stackrel{^}{p}}_{1}\left(1-{\stackrel{^}{p}}_{1}\right)}{{n}_{1}}+\frac{{\stackrel{^}{p}}_{2}\left(1-{\stackrel{^}{p}}_{2}\right)}{{n}_{2}}}$

$=\left(0.5498-0.4481\right)-1.96×\sqrt{\frac{0.5498\left(1-0.5498\right)}{522}+\frac{0.4481\left(1-0.4481\right)}{520}}$

$\approx -0.0191$

and $\left({\stackrel{^}{p}}_{1}-{\stackrel{^}{p}}_{2}\right)+{z}_{\alpha /2}×\sqrt{\frac{{\stackrel{^}{p}}_{1}\left(1-{\stackrel{^}{p}}_{1}\right)}{{n}_{1}}+\frac{{\stackrel{^}{p}}_{2}\left(1-{\stackrel{^}{p}}_{2}\right)}{{n}_{2}}}$

$=\left(0.5498-0.4481\right)+1.96×\sqrt{\frac{0.5498\left(1-0.5498\right)}{522}+\frac{0.4481\left(1-0.4481\right)}{520}}$

$\approx 0.1017$

The confidence interval is $\left(-0.0191,0.1017\right)$

## Step 3: To check if there is evidence of difference in true proportion of US men and women who can identify Egypt on map or not.

As the confidence interval $\left(-0.0191,0.1017\right)$ does not contain zero. It is unlikely that population proportions are equal.

So, there is evidence of difference in true proportion of US men and women who can identify Egypt on map or not. ### Want to see more solutions like these? 