The Choice of Where to Build Nests Is an Important $p = 0.30$

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