Self-Check Questions (ungraded)

CLT

Aidong Adam Ding
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Question 1:

\({X_1},\;...,\;{X_{100}}\) are independent random samples from the F-distribution with degrees of freedom 3 and 12.

  1. What is the probability that the sample mean \(\bar X\) is greater than 1.5?

Use the F-distribution formula to find the mean and variance for each random variable.

Then apply the Central Limit Theorem (CLT) to calculate the probability.

  1. 0.0082

This is calculated with the following R commands

F.mean<-12/(12-2)

F.var<- 2*12^2*(3+12-2)/3/10^2/8

1-pnorm(1.5,mean=F.mean,sd=sqrt(F.var/100))

Question 2:

The gene CCND3's expression value follows a normal distribution with mean 1.9 and standard deviation 0.5.

  1. What is the probability that the average of 33 measures of CCND3's expression value is between 1.5 and 1.8?[keep 3 decimal places]

We are NOT asking the probability about one gene expression value.

Rather the probability is on the sample mean of 33 expression values. Apply the Central Limit Theorem (CLT) to calculate the probability.

  1. 0.1253

This is calculated with the following R commands

pnorm(1.8, mean=1.9, sd=0.5/sqrt(33))-pnorm(1.5, mean=1.9, sd=0.5/sqrt(33))