Question

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} )?
the sampling distribution of ( hat{p} ) is approximately normal because ( np = 17.4>10 ).
because ( 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.
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.
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 right.

Explanation:

To determine the shape of the sampling distribution of \(\hat{p}\), we use the following rule: The sampling distribution of \(\hat{p}\) is approximately normal if \(np\geq10\) and \(n(1 - p)\geq10\). Here, \(n = 20\) and \(p=0.87\).

• Calculate \(np=20\times0.87 = 17.4\)
• Calculate \(n(1 - p)=20\times(1 - 0.87)=20\times0.13 = 2.6\)

Since \(n(1 - p)=2.6<10\), the sampling distribution of \(\hat{p}\) is not approximately normal.
When \(p\) is close to \(1\) (in this case \(p = 0.87\)), the sampling distribution of \(\hat{p}\) is skewed to the left. This is because there is a limited amount of values that \(\hat{p}\) can take on the lower - end (since \(p\) is large, the number of non - popcorn eaters in the sample is small).

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.