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### Course: AP®︎/College Statistics>Unit 9

Lesson 5: Sampling distributions for differences in sample proportions

# Mean and standard deviation of difference of sample proportions

## Problem

An educational researcher is studying two similar high schools in different states. Suppose that $50\mathrm{%}$ of all $425$ students at high school A have taken a college-level course, while $40\mathrm{%}$ of all $525$ students from high school B have taken such a course. The researcher plans on taking separate random samples of $50$ students from each high school to look at the difference $\left(\text{A}-\text{B}\right)$ between the proportions of students who have taken a college-level course in each sample.
Consider the formula:
${\sigma }_{{\stackrel{^}{p}}_{\text{A}}-{\stackrel{^}{p}}_{\text{B}}}=\sqrt{\frac{{p}_{\text{A}}\left(1-{p}_{\text{A}}\right)}{{n}_{\text{A}}}+\frac{{p}_{\text{B}}\left(1-{p}_{\text{B}}\right)}{{n}_{\text{B}}}}$
Why is it not appropriate for the researcher to use this formula for the standard deviation of ${\stackrel{^}{p}}_{\text{A}}-{\stackrel{^}{p}}_{\text{B}}$?