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ECON 317 Assignment 3

Instructions

This assignment contains 6 questions and must be completed with R. Write down your answers and codes in a R Markdown file (.rmd) using the RStudio, then generate a PDF file. Upload the R Markdown file and the PDF le to Canvas.

Question 1 [4 points]

Dice Rolling Experiment

A pair of fair dice are rolled 5 times. Let X be a random variable equal to the number of times the up faces on the two dice sum to either 7 or 11. Write an R script that will produce the PMF and CDF of X, (i.e., a table with X = x, f(x), and F(x) as columns) and plot the PMF of X.

Question 2 [9 points]

Generate 10 random numbers from a uniform distribution on [0,10]. Also, find the maximum and minimum values.

(a) For normal (0,1), find a number z* solving P (Z ≤ z*) = .05.

(b) For normal (0,1), find a number z* solving P (−z* ≤ Z ≤ z*) = .05

(c) How much area (probability) is to the right of 1.5 for a normal (0,2)?

Question 3 [8 points]

The distribution of a professor’s exam scores is normally distributed with a mean of 60 points and a standard deviation of 15 points. The professor promises his students that the top 20% of the scores will receive a grade of A in his class.

(a) What is the minimum score you must achieve to receive an A in the class?

(b) Sketch the graph and shade the concerned region.

Question 4 [5 points]

Initialize a vector of 100 zeros called sample_means. Run a loop that takes a sample of size 50 from N(30, 49) and stores the sample mean in sample_means, but only iterate from 1 to 100. Print the output to your screen. How many elements are there in this object called sample_means?

Question 5 [9 points]

Let  be the mean of a random sample of size 50 drawn from a population with mean 112 and standard deviation 40.

(a) Find the mean and standard deviation of .

(b) Find the probability that  assumes a value between 110 and 114.

(c) Find the probability that  assumes a value greater than 113.

Question 6 [15 points]

A random sample of size 30 is obtained from a normally distributed population with population mean 500 and a standard deviation of 100

(a) Construct a sampling distribution of the mean that has 1000 samples

(b) Calculate its Mean, Variance and Standard Error.

(c) Make a plot that shows the histograms for each of your 1000 samples

(d) using 3 different sample sizes (10, 100, 500), make three different histograms with normal curve. what happens to the shape of the histogram with increasing sample size?

(e) Describe the shape of this sampling distribution and compare the sampling distribution for all the sample sizes.

(f) What is the probability of getting a mean of 460?