The Choice of Where to Build Nests Is an Important $\hat{p}$

The choice of where to build nests is an important one for small bats. Their survivaland reproduction depends on making good choices. Research has suggested that thewarmth of the nest location is important, as well as protection from predators. In astudy of bats nesting under bridges over one and two lane roads in Louisiana,researchers investigated the theory that nest location is affected by danger frompredators. The researchers reasoned that if bats were seeking to avoid predators theywould roost in the narrowest spaces available under the bridge.
a) Forty-seven nests of a particular species were located under bridges on one-laneroads. Of these 47 nests, 28 of them were under the narrowest spaces of thebridges. Determine whether or not the sample size is large enough to use thestandard normal distribution to find the margin of error. Be sure to show anysupporting calculations.When this species of bat nested under bridges on two-lane roads, 24 out of 38 nests werefound in the narrowest spaces.
b) Assuming these nests to be representative of nests of this species under two-lanebridges, estimate the population proportion of nests that are in the narrowest ofspaces.
c) On the basis of prior observations and theory, the researchers believed that theproportion of nests located in the narrowest spaces is p = 0.30. If their belief iscorrect, what is the standard error of $\hat{p}$ ?
d) Calculate a 95% confidence interval for p, the proportion of the nests of thisspecies choosing the narrowest spaces when nesting under two-lane bridges.
e) Suppose the researchers wished to estimate the proportion of nests of this speciesthat would be in the narrowest spaces of bridges when under two-lane roads.What sample size would be needed to obtain a margin of error of 0.05 with 95%confidence?