Question

# American-made cars Nathan and Kyle both work for the Department of Motor Vehicles (DMV), but they live in different states. In Nathan's state, 80% of the registered cars are made by American manufactures. In Kyle's state, only 60% of the registered cars are made by American manufacturers. Nathan selects a random sample of 100 cars in his state and Kyle selects a random sample of 70 cars in his state. Let $\hat p_N - \hat p_K$ be the difference (Nathan's state-Kyle's state) in the sample proportion of cars made by American manufacturers. Calculate and interpret the standard deviation of the sampling distribution.

Solution

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Given:

\begin{align*} n_1&=\text{Sample size}=100 \\ n_2&=\text{Sample size}=70 \\ p_1&=80\%=0.80 \\ p_2&=60\%=0.60 \end{align*}

The mean of the sampling distribution of $\hat{p}_1-\hat{p}_2$ is the difference between the proportions $p_1$ and $p_2$.

$\mu_{\hat{p}_1-\hat{p}_2}=p_1-p_2=0.80-0.60=0.20$

Since the sample of 100 cars is less than 10% of all cars in Nathan's state and the sample of 70 cars are less than 10% of all cars in Kyle's state, the standard deviation of the sampling distribution of $\hat{p}_1-\hat{p}_2$ is given by $\sqrt{\dfrac{p_1(1-p_1)}{n_1}+\dfrac{p_2(1-p_2)}{n_2}}$.

\begin{align*} \sigma_{\hat{p}_1-\hat{p}_2}&=\sqrt{\dfrac{p_1(1-p_1)}{n_1}+\dfrac{p_2(1-p_2)}{n_2}} \\ &=\sqrt{\dfrac{0.80(1-0.80)}{100}+\dfrac{0.60(1-0.60)}{70}} \\ &=\sqrt{\dfrac{0.80(0.20)}{100}+\dfrac{0.60(0.40)}{70}} \\ &\approx 0.0709 \end{align*}

The difference (Nathan's state - Kyle's state) in the sample proportion of cars made by American manufacturers varies on average by 0.0709 from the mean difference 0.20.

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