Using R script functions generate random samples and calculate sample means. Be sure to solve all 3 parts of this problem.

Question

Please show the R script coding/steps to solve this. Please do all 3 parts of this question!!! Thanks in advance!!! :)

Assume that we have a normal population with a mean of 10 and a variance of 22, i.e., X∼N(10,22). In R, consider using the rnorm()function that is used to draw random values from a normal distribution (so it starts with ‘r’). It can be implemented as rnorm(n, mean, sd) where n is the number of random values to draw, and mean and sd are the mean and the standard deviation of the normal distribution from which random values are drawn.

Part 1: By using the rnorm()function, generate a random sample of size 1500 from the population and draw a histogram of the randomly drawn values.

Part 2: From the population, draw 50 random samples of size 30. Calculate the sample mean of each random sample generating 50 sample means from 50 random samples. Draw a histogram of the sample means.

Part 3: Compare the distributions of the population and the sample mean in terms of location and variability (by comparing the histograms). For this, calculate the mean and standard deviation of 1500 X’s in part 1 and the mean and standard deviation of 50 X’s in part 2. Determine if there is any significant difference in location or variability between the two distributions. If there is a difference, why do you see such a difference (think about the mathematical expressions)?

Answer 1

Sounds like a homework problem or an exam. Well, here are the parts needed to answer it. The student can then use these parts to get the final answer.

1. a) format for random normal function is: rnorm(n, mean = 0, sd = 1) where n is number of numbers to be generated. mean is the mean for the sample, and sd = standard deviation

b) basic format for histogram is: hist(x) where x = vector with numbers in it

the following example would generate 1000 samples from a population of mean 20 and standard deviation of 5 then plot a histogram

x<-rnorm(1000,20,5)

hist(x)

2,a) the 'for loop' has the following format : for ( i in b:e) { executable code }

where i is the index which increments from b to e

b) the sample function will pull samples from a population and put them in a named variable. its format is:

sample(x, size, replace = FALSE). Where x is name of variable/vector with values to sample. size is the number of samples to take, replace = indicates if the sampling will be done with or without replacement.

c). the mean function will generate the mean of the values in a vector variable. The format is:

mean(x)

d) to create a vector of a series of means, you will use the concatenate function c(x1,x2,...x4)

an example of code that would use these to put together a vector with the mean of 25 random samples is:

(note: since it wasn't specified , we do sampling with replacement to maintain population)

x1 = NULL

for (i in 1:25) {

s = sample(x,40, replace=TRUE)

m = mean(s)

x1<- c(x1,m)

}

hist(x1)

3 For this question the new function you will need will be standard deveation. sd().

sd(x). where x is vector variable. calculates standard deviation of the values in x

Remember , if you followed the above examples you would have two vectors of numbers. X. is a vector of 1500 random numbers. X1 is a vector of the MEANS of samples of X. The standard deviation of X would be the standard deviation of the population. The standard deviation of X1 would be the standard error of the sample means. They will vary. an example of how to examine the mean and standard deviation and distribution of these two different vectors is as follows

mean(x)

sd(x)

mean(x1)

sd(x1)

hist(x)

hist(x1)