the proportion of all us adults who eat popcorn when they go to the movie theater is $p = 0.87$. a random…

the proportion of all us adults who eat popcorn when they go to the movie theater is $p = 0.87$. a random sample of 20 us adults was selected and asked if they eat popcorn when they go to the movie theater. which of the following is the shape of the sampling distribution of $hat{p}$?\n\nthe sampling distribution of $hat{p}$ is approximately normal because $np = 17.4>10$.\n\nbecause $n(1 - p)=20(1 - 0.87)=2.6 < 10$, the sampling distribution of $hat{p}$ is not approximately normal. the sampling distribution of $hat{p}$ is skewed right and centered at 0.87.\n\nbecause $n(1 - p)=20(1 - 0.87)=2.6 < 10$, the sampling distribution of $hat{p}$ is not approximately normal. because $p = 0.87$ is closer to 1 than 0, the sampling distribution of $hat{p}$ is skewed to the left.\n\nbecause $n(1 - p)=20(1 - 0.87)=2.6 < 10$, the sampling distribution of $hat{p}$ is not approximately normal. because $p = 0.87$ is closer to 1 than 0, the sampling distribution of $hat{p}$ is skewed to the right.

the proportion of all us adults who eat popcorn when they go to the movie theater is $p = 0.87$. a random sample of 20 us adults was selected and asked if they eat popcorn when they go to the movie theater. which of the following is the shape of the sampling distribution of $hat{p}$?\n\nthe sampling distribution of $hat{p}$ is approximately normal because $np = 17.4>10$.\n\nbecause $n(1 - p)=20(1 - 0.87)=2.6 < 10$, the sampling distribution of $hat{p}$ is not approximately normal. the sampling distribution of $hat{p}$ is skewed right and centered at 0.87.\n\nbecause $n(1 - p)=20(1 - 0.87)=2.6 < 10$, the sampling distribution of $hat{p}$ is not approximately normal. because $p = 0.87$ is closer to 1 than 0, the sampling distribution of $hat{p}$ is skewed to the left.\n\nbecause $n(1 - p)=20(1 - 0.87)=2.6 < 10$, the sampling distribution of $hat{p}$ is not approximately normal. because $p = 0.87$ is closer to 1 than 0, the sampling distribution of $hat{p}$ is skewed to the right.

Answer

Explanation:

Step1: Check normal - approximation conditions

For the sampling distribution of $\hat{p}$ to be approximately normal, we need $np\geq10$ and $n(1 - p)\geq10$. Given $n = 20$ and $p=0.87$, we calculate $np=20\times0.87 = 17.4\geq10$ and $n(1 - p)=20\times(1 - 0.87)=20\times0.13 = 2.6<10$. So the sampling distribution is not approximately normal.

Step2: Determine the skewness

When $p$ is close to 1, the sampling distribution of $\hat{p}$ is skewed to the left. Since $p = 0.87$ is close to 1, the sampling distribution of $\hat{p}$ is skewed to the left.

Answer:

Because $n(1 - p)=20(1 - 0.87)=2.6 < 10$, the sampling distribution of $\hat{p}$ is not approximately Normal. Because $p = 0.87$ is closer to 1 than 0, the sampling distribution of $\hat{p}$ is skewed to the left.