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.

Question

Quizlet Admin · Accepted Answer

Given:

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

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.

Related questions with answers

Create an account to view solutions

Create an account to view solutions

Recommended textbook solutions

The Practice of Statistics for the AP Exam

Updated The Practice of Statistics for the AP Exam

The Practice of Statistics for AP

The Practice of Statistics for the AP Exam

More related questions