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Q 10.

Expert-verified

The Practice of Statistics for AP Examination

Found in: Page 640

Book edition 6th

Author(s) Daren Starnes, Josh Tabor

Pages 837 pages

ISBN 9781319113339

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Short Answer

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 true
proportion 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 true
proportions of U.S. men and women who can identify Egypt on a map? Justify your
answer.

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

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

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

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: ${\hat{p}}_{1} = \frac{x_{1}}{n_{1}} = \frac{287}{522} = 0.5498$

${\hat{p}}_{2} = \frac{x_{2}}{n_{2}} = \frac{233}{520} = 0.4481$

Confidence Interval:

$({\hat{p}}_{1} - {\hat{p}}_{2}) - z_{α / 2} \times \sqrt{\frac{{\hat{p}}_{1} (1 - {\hat{p}}_{1})}{n_{1}} + \frac{{\hat{p}}_{2} (1 - {\hat{p}}_{2})}{n_{2}}}$

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

$\approx - 0.0191$

and $({\hat{p}}_{1} - {\hat{p}}_{2}) + z_{α / 2} \times \sqrt{\frac{{\hat{p}}_{1} (1 - {\hat{p}}_{1})}{n_{1}} + \frac{{\hat{p}}_{2} (1 - {\hat{p}}_{2})}{n_{2}}}$

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

$\approx 0.1017$

The confidence interval is $(- 0.0191, 0.1017)$

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 $(- 0.0191, 0.1017)$ 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.

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