Question

Generate 20 samples of size n = 30 from your population. To do this: i.

Generating a sample of just size n = 30. Calculate the sample mean, X¯, for this sample. Record this value somewhere in your spreadsheet (you will need it later).

ii. Repeat the previous step 19 more times, so that you end up with a spreadsheet with 20 columns, each column has 30 randomly generated values from your population, and you have calculated a sample mean for each of your 20 samples of n = 30 observations.

(i) Create a histogram of the 20 sample means you have calculated, making sure your histogram has appropriate axis labels and a title. Include this histogram in your report, along with a commentary on what this distribution represents. Does the shape of this distribution agree with your prediction in part

(g)?. NOTE: You may need to adjust the number of bars/bins in your histogram to get a better representation of the shape of the data set.

(j) Calculate the mean and standard deviation of the 20 sample means you have calculated. Include these values in your report. How closely do these values match what you predicted in part (g)? (k) Provide a short (i.e. 2 - 3 sentence) summary of what exactly it is you just demonstrated in this question.

Answer 1

a)Generating a sample of just size n = 30. Calculate the sample mean

We do this procedure in R

sample20<-data.frame(replicate(20,rnorm(30,mean=50,sd=6)))

Here we are generating first a sample of size 30 with mean = 50 and standard deviation = 6.

Then using the replicate function we are replicating the same 20 times.

Here is a view of the data:

b)Repeat the previous step 19 more times, so that you end up with a spreadsheet with 20 columns, each column has 30 randomly generated values from your population, and you have calculated a sample mean for each of your 20 samples of n = 30 observations.

The sample means for each of 20 samples of n = 30 observations is calculated by the function

colMeans(sample20) # sample20 is the name of the data as we have 20 samples.

(c)

Create a histogram of the 20 sample means you have calculated, making sure your histogram has appropriate axis labels and a title. Include this histogram in your report, along with a commentary on what this distribution represents. Does the shape of this distribution agree with your prediction in part

hist(colMeans(sample20))

d)

Calculate the mean and standard deviation of the 20 sample means you have calculated. Include these values in your report. How closely do these values match what you predicted in part (g)?

> mean(colMeans(sample20))
[1] 49.90496
> sd(colMeans(sample20))
[1] 1.300241​

(k) Provide a short (i.e. 2 - 3 sentence) summary of what exactly it is you just demonstrated in this question.

We simply created a sample of size 30 randomly distributed with mean 50 and sd 6. Then we replicated this thing 20 times.

We got a dataset of size 20=30= 600 data points.

Then we calculated the means of each of the 20 columns and created a histogram of them. They were around 50 the actual mean.

Then we calculated the mean of means of these 20 columns which was very close to the actual mean. and standard deviation of the means which was close to the predicted value of

$s/\sqrt{n}=6/\sqr{20}=1.34$