Question

In the notes there is a Central Limit Theorem example in which a sampling distribution of means is created using a for loop, and then this distribution is plotted. This distribution should look approximately like a normal distribution. However, not all statistics have normal sampling distributions. For this problem, you'll create a sampling distribution of standard deviations rather than means. 3. Using a for loop, draw 10,000 samples of size n-30 from a uniform distribution with min-100 and max-300. For each sample, compute the standard deviation. Make a histogram of the distribution of standard deviations. Give your plot appropriate labels Repeat the procedure in a., but this time draw samples of n-50. Repeat the procedure in a, but this time draw from a uniform distribution with min-100 and max=500. Redo the plots from a, b, and c, this time making the x and y axes for each plot have the same scale, so that they are easy to compare to one another. What effect did changing the sample size have on the sampling distribution of the standard deviation? What effect did changing the max value for the uniform distribution that the samples were drawn from have on the sampling distribution of the standard deviation? a. b. c. d.

R Programming codes for the above questions?

Answer 1

Answer:

a)  

sdx <- rep(0,10000)
set.seed(10)
for (i in 1:10000) {
x <- runif(30,100,300)
sdx[i]=sd(x)
}

hist(sdx, probability = TRUE, xlab = "Sample standard deviation")

b)   

sdx <- rep(0,10000)

set.seed(10)
for (i in 1:10000) {
x <- runif(50,100,300)
sdx[i]=sd(x)
}
hist(sdx, probability = TRUE, xlab = "Sample standard deviation")

c)

sdx <- rep(0,10000)

set.seed(10)
for (i in 1:10000) {
x <- runif(30,100,500)
sdx[i]=sd(x)
}
hist(sdx,probability = TRUE,xlab = "Sample standard deviation")

d) For multiple histograms to compare, we will use 'par' command. and For common scales, we will set a common limit on x axis, i.e. 0 to 150.

par(mfrow=c(2,2))

sdx <- rep(0,10000)
set.seed(10)
for (i in 1:10000) {
x <- runif(30,100,300)
sdx[i]=sd(x)
}

hist(sdx,probability = TRUE,xlab = "Sample standard deviation",xlim = c(0,150))

sdx <- rep(0,10000)
set.seed(10)
for (i in 1:10000) {
x <- runif(50,100,300)
sdx[i]=sd(x)
}
hist(sdx,probability = TRUE,xlab = "Sample standard deviation",xlim = c(0,150))

set.seed(10)
sdx <- rep(0,10000)
for (i in 1:10000) {
x <- runif(30,100,500)
sdx[i]=sd(x)
}
hist(sdx,probability = TRUE,xlab = "Sample standard deviation",xlim = c(0,150))

Histogram of sdx Histogram of sdx 50 100 150 50 100 150 Sample standard deviation Sample standard deviation Histogram of sdx 50 100 150 Sample standard deviation

The histogram squeezes as we increase the sample size and becomes more normally distributed.

On changing the maximum value for uniform distribution, the graph shifts towards the right.

  For any query related to the codes used above, please comment the same.