`III. SAMPLING DISTRIBUTIONS`

`A. SAMPLE MEAN`

`1. POPULATION VARIANCE KNOWN`

`a. NORMAL POPULATION`

`The mean of, when x _{i} 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 s ^{2}, 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 s ^{2}
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 s ^{2} `

`Assume `

`Recall`

`where`

`or`

`Note`

`recall `

`Also`

`We therefore conclude the following`

*Example:* Find s^{2} when n = 17, = 40

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

`III.B.2. DISTRIBUTION OF s _{1}^{2}/s_{2}^{2}`

`Assume`

`Recall`