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Constructing a $2$ Proportion $z$-Interval: High School Break Up

APSTAT-VLXVWY

Independent random samples of $50$ boys and $50$ girls in a school were asked if they are depressed for more than a week after they got broken up with by their significant other. Of the $50$ boys, $26$ said yes, and of the $50$ girls, $35$ said yes.

Which of the following is a $95\%$ confidence interval for the difference in the proportion of boys compared to the proportion of girls who are depressed for more than a week after they have been broken with by their significant other in this school?

A

$(0.52-0.70)\pm 1.645\sqrt { \cfrac { (0.52)(0.48) }{ 50 } +\cfrac { (0.70)(0.30) }{ 50 } }$

B

$(0.52-0.70)\pm 1.96\sqrt { \cfrac { (0.52)(0.48) }{ 50 } +\cfrac { (0.70)(0.30) }{ 50 } }$

C

$(0.52-0.70)\pm 1.645\sqrt { (\cfrac { 61 }{ 100 } )(\cfrac { 39 }{ 100 } )(\cfrac { 1 }{ 50 } +\cfrac { 1 }{ 50 } ) }$

D

$(0.52-0.70)\pm 1.96\sqrt { (\cfrac { 61 }{ 100 } )(\cfrac { 39 }{ 100 } )(\cfrac { 1 }{ 50 } +\cfrac { 1 }{ 50 } ) }$

E

$(0.52-0.70)\pm 1.96\sqrt { (\cfrac { 26 }{ 100 } )(\cfrac { 35 }{ 100 } )(\cfrac { 1 }{ 50 } +\cfrac { 1 }{ 50 } ) }$