Description

Hanao Li
Instructions
Before you leave lab today make sure that you upload an RMarkdown file to the canvas page (this should have a .Rmd extension) as well as the html output after you have knitted the file (this will have a .html extension). Note that since you have already knitted this file, you should see both a Lab1_UNI.html and a Lab1_UNI.Rmd file in your GR5206 folder. Click on the Files tab to the right to see this. The files you upload to the Canvas page should be updated with commands you provide to answer each of the questions below. You can edit this file directly to produce your final solutions.
Background: The Normal Distribution
Recall from your probability class that a random variable X is normally-distributed with mean μ and variance σ2
(denoted X ∼ N(μ,σ2)) if it has a probability density function, or pdf, equal to
f .
In R we can simulate N(μ,σ2) random variables using the function. For example,

outputs 5 normally-distributed random variables with mean equal to 10 and standard deviation (this is σ) equal to 3. If the second and third arguments are ommited the default rates are mean = 0 and sd = 1, which is referred to as the “standard normal distribution”.
Tasks
Sample means as sample size increases
1. Generate 100 random draws from the standard normal distribution and save them in a vector named normal100. Calculate the mean and standard deviation of normal100. In words explain why these values aren’t exactly equal to 0 and 1.

They values aren’t exactly eqaul to 0 and 1 because these draws are randomly generated from the standard normal distributions, so there will be some discrapencies. When the sample is large enough, the value then will be very close to 0 and 1.
2. The function hist() is a base R graphing function that plots a histogram of its input. Use hist() with your vector of standard normal random variables from question (1) to produce a histogram of the standard normal distribution. Remember that typing ?hist in your console will provide help documents for the hist() function. If coded properly, these plots will be automatically embedded in your output file.

3. Repeat question (1) except change the number of draws to 10, 1000, 10,000, and 100,000 storing the results in vectors called normal10, normal1000, normal10000, normal100000.

4. We want to compare the means of our four random draws. Create a vector called sample_means that has as its first element the mean of normal10, its second element the mean of normal100, its third element the mean of normal1000, its fourth element the mean of normal10000, and its fifth element the mean of normal100000. After you have created the sample_means vector, print the contents of the vector and use the length() function to find the length of this vector. (it should be five). There are, of course, multiple ways to create this vector. Finally, explain in words the pattern we are seeing with the means in the sample_means vector.

From those five sample means, we could see that when the samples become larger and larger, the mean of the samples will become more and more accurate and towards 0.
Sample distribution of the sample mean
5. Let’s push this a little farther. Generate 1 million random draws from a normal distribution with μ = 3 and 2
σ = 4 and save them in a vector named normal1mil. Calculate the mean and standard deviation of normal1mil.

7. Create a matrix normal1mil_mat from the vector normal1mil that has 10,000 columns (and therefore should have 100 rows).

8. Calculate the mean of the 1234th column.

9. Use the colSums() functions to calculate the means of each column of normal1mil_mat. Remember, ? colSums will give you help documents about this function. Save the vector of column means with an appropriate name as it will be used in the next task.

10. Finally, produce a histogram of the column means you calculated in task (9). What is the distribution that this histogram approximates (i.e. what is the distribution of the sample mean in this case)?

The distribution is similar to the normal distribution.