(Solved): A researcher believes that \( 51 \% \) of people who grew up as the only child have an IQ score ov ...

A researcher believes that \( 51 \% \) 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 \( \hat{p} \) be the proportion of people in the sample with an IQ score above 100. Answer the following. (If necessary, consult a list of formulas.) (a) Find the mean of \( \hat{p} \). (b) Find the standard deviation of \( \hat{p} \). (c) Compute an approximation for \( P(\hat{p} \geq 0.51) \), which is the probability that there will be \( 51 \% \) or more people with IQ scores over 100 in the sample. Round your answer to four decimal places.