The rare booted eagle of western Europe was the focus of a study by S. Suarez et al. to identify optimal nesting habitat for this raptor. According to their paper "Nesting Habitat Selection by Booted Eagles (Hieraaetus pennatus) and Implications for Management" (Journal of Applied Ecology, Vol. 37, pp. 215-223), the distances of such nests to the nearest marshland are normally distributed with mean 4.66 km and standard deviation 0.75 km.
a. Simulate 3000 samples of four distances each.
b. Determine the sample mean and sample standard deviation of each of the 3000 samples.
c. For each of the 3000 samples, determine the observed value of the standardized version of x-bar.
d. Obtain a histogram of the 3000 observations in part (c).
e. Theoretically, what is the distribution of the standardized version of x-bar?
f. Compare your results from parts (d) and (e).
g. For each of the 3000 samples, determine the observed value of the studentized version of x-bar.
h. Obtain a histogram of the 3000 observations in part (g).
i. Theoretically, what is the distribution of the studentized version of x-bar?
j. Compare your results from parts (h) and (i).
k. Compare your histograms from parts (d) and (h). How and why do they differ?