A researcher believes that 52% of people who grew up as the only child have an IQ score over 100. However, unknown to the researcher, this figure is actually 50% , which is the same as in the general population. To attempt to find evidence for the claim, the researcher is going to take a random sample of 400 people who grew up as the only child. Let p-hat be the proportion of people in the sample with an IQ score above 100.

Find the mean of P-hat

Find the Standard Divination of P-hat

Compute an approximation for P(p-hat greater than of equal to 0.52) which is the probability that there will be 52% or more people with IQ scores over 100 in the sample. Round your answer to four decimal places.