## Related questions with answers

Your teacher brings two large bags of colored goldfish crackers to class. Bag 1 has 25% red crackers and Bag 2 has 35% red crackers. Using a paper cup, your teacher takes an SRS of 50 crackers from Bag 1 and a separate SRS of 40 crackers from Bag 2. Let $\hat{p}_{1}-\hat{p}_{2}$ be the difference in the sample proportions of red crackers. What is the shape of the sampling distribution of $\hat{p}_{1}-\hat{p}_{2} ?$ Why?

Solution

VerifiedGiven:

$\begin{align*} n_1&=\text{Sample size}=50 \\ n_2&=\text{Sample size}=40 \\ p_1&=25\%=0.25 \\ p_2&=35\%=0.35 \end{align*}$

It is safe to assume that the sampling distribution of $\hat{p}_1-\hat{p}_2$ is approximately normal if $n_1p_1\geq 10$, $n_1(1-p_1)\geq 10$, $n_2p_2\geq 10$ and $n_2(1-p_2)\geq 10$.

$\begin{align*} n_1p_1&=50(0.25)=12.5\color{#4257b2}\geq 10 \\ n_1(1-p_1)&=50(1-0.25)=50(0.75)=37.5\color{#4257b2}\geq 10 \\ n_2p_2&=40(0.35)=14\color{#4257b2}\geq 10 \\ n_2(1-p_2)&=40(1-0.35)=40(0.65)=26\color{#4257b2}\geq 10 \end{align*}$

We note that $n_1p_1\geq 10$, $n_1(1-p_1)\geq 10$, $n_2p_2\geq 10$ and $n_2(1-p_2)\geq 10$,which implies that the sampling distribution of $\hat{p}_1-\hat{p}_2$ is approximately normal.

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