The First Large-Scale Study of the Human Sex Ratio Involved $( \hat { p } = 0.60$

The first large-scale study of the human sex ratio involved over 72,000 children in 19th Century Germany. Fifty-two percent of the children in the study were boys. Suppose that 21 st Century researchers wish to replicate this observational study to see if the proportion of boys might have changed in the intervening years. A random sample of 50 children included 30 boys $( \hat { p } = 0.60$ .)
a) If the modern true population proportion of newborn boys is $p = 0.52$ , show that it is reasonable to approximate the sampling distribution of $\hat { p }$ for random samples of size 50 using a normal distribution.
b) If the modern true population proportion of newborn boys is $p = 0.52$ , what is the probability of observing a sample proportion of newborn boys that is greater than $\hat { p } = 0.6$ for a random sample of size 50 ? c) If the modern true population proportion of newborn boys is $p = 0.52$ , what is the probability of observing a sample proportion of newborn boys that is less than $\hat { p } = 0.5$ for a random sample of size $50 ?$