Question

According to a poll, 55 % of Americans do not know that GOP stands for Grand Old Party (Time, October 17, 2011). Assume that this percentage is true for the current population of Americans. Let p ^ be the proportion in a random sample of 953 Americans who do not know that GOP stands for Grand Old Party. Find the mean and standard deviation of the sampling distribution of p ^ and describe its shape.

Answer 1

`\hat p  \sim N (p , \sqrt{(p(1-p))/(n)})`

The mean is given by:

`\mu_(\hat p) = 0.55`

And the deviation:

`\sigma_(\hat p) =\sqrt{(0.55*(1-0.55))/(953)}= 0.0161`

Explanation:

For this case we assume that the true population proportion of Americans do not know that GOP stands for Grand Old Party is 0.55 and we select a random sample of n = 953 americans

For this case we assume that we satisfy the conditions to use the normal approximation for
`\hat p`

1) np >10 , n(1-p)>10

2) Independence

3) Random sample

4) The sample size is less than 10% of the population size

We assume that all the conditions are satisfied and the distribution for
`\hat p` would be:

`\hat p  \sim N (p , \sqrt{(p(1-p))/(n)})`

The mean is given by:

`\mu_(\hat p) = 0.55`

And the deviation:

`\sigma_(\hat p) =\sqrt{(0.55*(1-0.55))/(953)}= 0.0161`