Suppose that an investigator hypothesized that consumers cannot really tell the difference between $2$ versions of a cola drink that are identical except one naturally contains caffeine and the other does not. If the investigator is correct, then if we give a person a sample of each cola, tell them that one of the $2$ identical$-$looking samples has contains caffeine and the other does not and ask them which one they think has the caffeine, they should correctly identify the one containing caffeine with probability $1/2.$

We will assume that the investigator's hypothesis is correct, meaning that the population proportion really is $\mathrm{.}5$ for this question.

The investigator carries out this experiment with n $=250$ subjects.

Question:

$1.$Explain why we can assume the sampling distrution of p$^$ is approximately normal?

$2.$what is the expected value of p$^?$

$3.$What percent of all possible samples will give p$^$ values between $\mathrm{.}45$ and $\mathrm{.}55?$