Let X be the mean of a random sample of size 100 from a rand

Let X be the mean of a random sample of size 100 from a random variable that is distributed with

mean 200 and variance3 400, and a distribution that is not normal.

a. What is the mean of X ?

b. What is the variance and standard deviation of X ?

c. Is X likely to be normally distributed? Explain.

Solution

a)

By the central limit theorem, even though the distirbution of the numbers themselves is not normal, the distribution of of the means will be approximately normal, with the same mean:

Mean = 200 [ANSWER]

*********************

b)

With variance, from central limit theorem still,

Variance of the means = variance / n = 3400/100 = 34

standard deviation of means = sqrt(34) = 5.830951895 [ANSWER]

*********************

c)

Yes, by central limit theorem, because there is a sufficiently large number of iterates, and a large sample size of 100.