Part C a

based on the histogram, the mode is around 25 (corresponding to r1) and the second peak is around 10 (corresponding to r2) as r1 and r2 have sd of 4 and 3, very few samples lie outside of the ranges (21, 29) and (7, 13). Because the sample size of r1 is twice that of r2, then frequency of 25 is higher than that of 10.

b Draw 50 samples of size 15, and plot the sampling distribution of means as a histogram.

c Draw 50 samples of size 45, and plot the sampling distribution of means as a histogram.

d Make sure that the distributions in parts b and c are on the same plot.

Similarities: three histograms reflect similar patterns because the modes are close to 20. Also, very few samples on both sides because they lie outside of one standard deviation from the mean.

Differences: the histogram from part a only contains 150 samples and the mode is not a unique number (three bars with approximately same heights). In contrast, histograms from part b & c only had one bar for the mode and is more likely to be normally distributed.

e

CLT: when adding independent random variables, we get a normal distribution of the samples. Samples represent the population more accurately when n increases. When several sets of samples merge together, the histogram reflects the sample means by frequencies. The pattern corresponds to the nature of the samples sets by means and sd.

f

This is very helpful to me because I can see how the increase of sample size influence the pattern of the histogram. The height of each bar can be compared directly with each other and the number of breaks on the x-axis can be managed to scrutinize the details of the samples.