Suppose that a sample of size n=3 is selected from a population of four value 4, 8, 0, 12. (1) Find the population distribution. (2) Find the mean and standard deviation of the population distribution. (3) Suppose that the sampling is without replacement. (a) Find the sampling distribution of the sample mean. (b) Find the mean of this sampling distribution. (c) Find the standard deviation of this sampling distribution. (Calculate standard deviation: https://www.youtube.com/watch?v=YMbt5rYzp-Q) (d) Compare the mean and standard deviation of this sampling distribution with those of the population. Summarize your findings. (e) Letp=the proportion of values in the population that equal O. Let be the proportion of values in the sample that equal O. (e1) Find the sampling distribution of the sample proportion. (e2) Find the mean of this sampling distribution. (e3) Find the standard deviation of this sampling distribution. (e4) Compare the mean of this sampling distribution with the population proportion p. Summarize your findings. (4) Suppose that the sampling is with replacement, (a) Find the sampling distribution of the sample mean. (b) Find the mean of this sampling distribution. (c) Find the standard deviation of this sampling distribution. (d) Compare the mean and standard deviation of this sampling distribution with those of the population. Summarize your findings. (e) Letp=the proportion of values in the population that equal O. Let p be the proportion of values in the sample that equal 0. (e1) Find the sampling distribution of the sample proportion. (e2) Find the mean of this sampling distribution. (e3) Find the standard deviation of this sampling distribution. (e4) Compare the mean of this sampling distribution with the population proportion p. Summarize your findings. (5) Suppose that the sample consists of the three numbers 4, 0, and 8. Now, treat this sample as a new population. Select, with replacement, three values from this new distribution and call this new sample a bootstrap sample. There are many different bootstrap samples. (a) What is the probability of getting each of the bootstrap samples? (b) Calculate the mean of each bootstrap sample. Then, all of these means, along with the probabilities just obtained, gives a distribution which can be used as an approximation to the sampling distribution of the sample mean. This distribution is called the bootstrap distribution of the sample mean. Create a table for it. (c) What is the mean of this bootstrap distribution? (d) Compare this mean with the sample mean (the mean of the original sample), what conclusion can you possibly draw? Remark: How many different bootstrap samples are possible? There is a general result we can use to count it: Given N distinct items, the number of ways of choosing n items with replacement from these items is given by N+n-1) . To count the number of bootstrap samples we discussed n above, we have N = 3 and n = 3. So, there are totally 3+3-1 -³-¹) = (3) = 10 bootstrap samples. 3 Submission: Submit an R Markdown that generates a neat html output. The html file should also be submitted. Don't use rpubs.com.