III. SAMPLING DISTRIBUTIONS

A. SAMPLE MEAN

1. POPULATION VARIANCE KNOWN

a. NORMAL POPULATION

The mean of, when xi are iid with is found by exploiting the fact that expectation is a linear operator

The var of is found by exploiting the same property

DISTRIBUTION OF

Suppose

The sample mean is a linear combination of random variables that are normally distributed. We therefore conclude that

also

Example: Consider the population of errors in statistics books. We assume that this r.v. is normally distributed. If we draw a sample of n = 9 from this population with= 20 and= 25. What is the probability thatwill exceed 22?

Note that as n increases, the variance ofdiminishes. Therefore

gets larger and the probability thatdiffers from the population mean diminishes. Recall the weak law of large numbers.

b. NON-NORMAL POPULATIONS

Example: A machine produces 100 sneakers at a time. Because of air bubbles, etc., there is a probability of .1 that a randomly selected sneaker is defective.

The mean number of defectives in a production run is

the variance is

Using the CLT we assert that x, the number of defectives in a run, is a normal random variable. Hence

If we were 'doubting Thomas' and did it using the binomial we would find

Which is a fair approximation?

The following flow diagram should aid you deciding when to use the binomial and when to rely on the normal as an approximation.

III. A.2. a. POPULATION VARIANCE UNKNOWN, NORMAL POPULATION

Consider the possibility that we do not know the population variance but do know that our random variable has a normal distribution. Fortunately we have s2, the sample variance, which might serve as a reasonable approximation of . Consequently, we construct the new random variable

Note: 1. The expected value of this is still zero.

2. We have added some uncertainty in using s2 to approximate. In fact, the variance depends on the sample size.

As a side note: Recall

and we can show

Let us divide theby its degrees of freedom

and take its square root

Now consider dividing our N(0, 1) by our

to get

After canceling terms and rearranging we get

Example: Acreage sales are normal. For any given year suppose that the mean acreage per sale in a particular state is known to be 100. We have calculated s to be 20 from a sample of n = 9. What is the probability that our sample mean will exceed 109?

III. A.2. b. NON NORMAL POPULATION

If the population is non-normal andis unknown then there is little that we can say, in a probabilistic sense, about the outcome of a sampling experiment. However, real life is not the same as theory and you will often see people using a t or normal probability table if the sample is reasonably large. This is wrong!

III.B.1. DISTRIBUTION OF s2

Assume

Recall

where

or

Note

recall

Also

We therefore conclude the following

Example: Find s2 when n = 17, = 40

From the chi square tables we find 'a' = 19.4

III.B.2. DISTRIBUTION OF s12/s22

Assume

Recall