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

Answer:

$\hat p \sim N (p , \sqrt{\frac{p(1-p)}{n}})$

The mean is given by:

$\mu_{\hat p} = 0.55$

And the deviation:

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

Step-by-step 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:

$\hat p \sim N (p , \sqrt{\frac{p(1-p)}{n}})$

The mean is given by:

$\mu_{\hat p} = 0.55$

And the deviation:

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