Problem Set 3

Problem Set Due Wednesday October 30 at 7pm on Canvas.

You will hand in a .Rmd file and a knitted html output.

I have provided the raw RMD of this problem set you can use as a template for answering.

Question 1

Women’s height in the US is normally distributed with a mean of 64.5 inches and a standard deviation of 2.5 inches. Answer the following questions based off that information.

  1. Take a random sample of 50 heights. Plot the distribution of the population and your sample of 50. How close are they? Is that what you expected?

  2. Taking advantage of the fact that I’ve given you information about the population, plot the sampling distribution of the sample mean for \(n=50\). Confirm that this sampling distribution is correct by repeatedly sampling from this distribution, taking means, and plotting the density of those simulated means.

  3. Calculate the probability that the mean height of a sample of 50 women will be less than 64 inches. Confirm your answer with the simulated results from (b).

  4. Calculate the probability that the mean height of a sample of women will be between 64 and 65 inches. Confirm your answer with the simulated results from (b).

  5. What is the probability that a woman is less than 64 inches tall? What is the probability a woman is between 64 and 65 inches tall? Give the answers, and then say in words why these answers are not the same as the answers for (c) and (d).

  6. What size \(n\) would you need to be 99% certain that the mean height in a sample of women will be between 64 and 65 inches. Determine this answer and then repeatedly sample from the population with a sample of that size to show that 99% of sample means do indeed fall within those bounds.

Question 2

Use the code below to load the data for this question. This file contains a single sample x from an unknown distribution.

library(rio)
x <- import("https://github.com/marctrussler/IIS-Data/raw/main/Question2.RDS", trust=T)

  1. Plot the distribution of this sample and determine the mean and variance. Using your intuition: what do you expect the distribution that produced this sample might have looked like? (Ignore that I tell you below, genuinely try to think about how you would answer this question if you didn’t know. )

  2. (Again ignoring that I tell you the population that this sample is drawn from below) Using only information contained in this one sample produce a 70%, 90% and 95% confidence interval around the mean of this sample. For each, describe in words what the confidence interval is telling you.

  3. The following code produced this sample: rnorm(10,56,32). Now using this information, repeatedly re-sample and calculate confidence intervals to show that 70%, 90% and 95% of similarly constructed confidence intervals contain the true population parameter.