Question

Part D: 1. Draw 500 random samples of size 8 from a random number generator from a standard normal distribution. Then increase the sample size to 32. Finally, increase the sample size to 128. Plot histograms of the sampling distributions of (i) the sample mean andi) the sample variance, for each of these three sample sizes. Now repeat your experiments for three samples drawn from another parametric distribution of your choice (e.g., a uniform distribution) Discuss the results of your experiments in light of the centra lit theorem 2. Your experiments produce "samples of sample means. Compute the mean and variance of the sample means generated by each experiment and compare them to the mean and variance predicted by statistical theory. Does the variance of the sample means (i.e., the sampling variance) decrease with the sample size at the rate predicted by the theory? Does Normality matter for this?

Answer 1

Draw 500 random samples of size 8 from a random number generator for a standard normal distribution. then increase the sample size to 32. Finally, increase the sample size to 128.

We given an R program for the simulation purpose. The 600 random samples is specified in "B <- 500" and the sample sizes of 8, 32, and 128 respectively through rnorm(8), rnorm(32) and rnorm(128).

> B <- 500
> bs8 = bs32 = bs128 <- NULL
> for(i in 1:B) {
+   bs8[i] <- mean(rnorm(8))
+   bs32[i] <- mean(rnorm(32))
+   bs128[i] <- mean(rnorm(128))
+ }
For each step, calculate the sample mean and the sample variance. 
> mean(bs8); mean(bs32); mean(bs128)
[1] -0.007853833
[1] 0.001011168
[1] -0.0021334
> var(bs8); var(bs32); var(bs128)
[1] 0.09262657
[1] 0.03790872
[1] 0.01066618
The simulation shows that as the sample size increases, the variability decreases.

Plot histograms of the sampling distributions of both estimator for each of these three sample sizes. 
> windows(height=15,width=5)
> par(mfrow=c(3,1))
> hist(bs8,main="CLT Based on Size 8")
> hist(bs32,main="CLT Based on Size 32")
> hist(bs128,main="CLT Based on Size 128")

CLT Based on Size 10 1.0 -0.5 0.0 0.5 10 bs10 CLT Based on Size 25 -0.6 -0.2 0.2 0.6 bs25 CLT Based on Size 100 -0.2 0.0 0.2 0.4 bs100

Repeat the same experiment with samples drawn from another distribution of your choice, for example an Uniform Distribution or a 2 -distribution. Discuss the results of these experiments in light of the CLT.

The normal distribution is replaced by a Gamma distribution and the previous R program is repeated all over again.

> B <- 500
> bsnonnormal08 = bsnonnormal32 = bsnonnormal128 <- NULL
> for(i in 1:B) {
+   bsnonnormal08[i] <- mean(rgamma(8,5,1.5))
+   bsnonnormal32[i] <- mean(rgamma(32,5,1.5))
+   bsnonnormal128[i] <- mean(rgamma(128,5,1.5))
+ }
> mean(bsnonnormal08); mean(bsnonnormal32); mean(bsnonnormal128)
[1] 3.309641
[1] 3.333957
[1] 3.332683
> var(bsnonnormal08); var(bsnonnormal32); var(bsnonnormal1228)
[1] 0.2112038
[1] 0.0860186
[1] 0.0228132
> windows(height=15,width=5)
> par(mfrow=c(3,1))
> hist(bsnonnormal08,main="CLT Based on Size 08")
> hist(bsnonnormal32,main="CLT Based on Size 32")
> hist(bsnonnormal128,main="CLT Based on Size 128")

CLT Based on Size 10 2.0 3.0 4.0 5.0 bsnonnormal10 CLT Based on Size 25 2.5 3.0 3.5 4.0 bsnonnormal25 CLT Based on Size 100 2.8 3.0 3.2 3.4 3.6 3.8 bsnonnormal100

The reduction is sample variance is slower now.

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