Sampling distribution of the sample mean

02/27/14

I will hand back exam 1 after class.
Exam solutions are now on Blackboard. Please double check why you lost points before you ask questions!
We're behind most the other sections, we've got lots of difficult material to cover!

Understand how to specify the sampling distribution of the sample mean (that is, how to find it's mean, variance and shape).
Understand how sampling distributions are created. The lecture notes uses the K-Packs example to demonstrate this idea. Think about exam scores.
This video nicely demonstrates the behavior of the sampling distribution when we increase sample size.

In Chapter 12, you found probabilities such as \( P(X < 5) \).
This process assumes the shape of the population distribution is normal!
We now relax this assumption – the shape can be either normal or non-normal.
For example, if the shape of the population distribution is non-normal, you cannot find \( P(X < 5) \)!

From now on, the notation \( X \sim N(\mu, \sigma^2) \) tells us 3 things about a population:
- Shape: Normal distribution
- Mean: \( \mu \)
- Standard Deviation: \( \sigma \)
If I write \( X \sim (\mu, \sigma^2) \), the shape is not normal.

In practice, \( \mu \) is unknown; so we use a sample mean to estimate it.
Since samples are random, the sample mean can be thought of as a random variable \( \bar{X} \) (before taking the sample).
All random variables have a distribution. The distribution of \( \bar{X} \) is called the sampling distribution of the sample mean.
Given the shape, mean, and variance of the population; it is fairly easy to specify the distribution of \( \bar{X} \).

Assuming we have a simple random sample, \( \bar{X} \) will be unbiased.
That is, the mean of \( \bar{X} \), denoted as \( (\mu_{\bar{X}}) \), is equal to…
the population mean \( (\mu) \).
The variance of \( \bar{X} \), denoted as \( \sigma^2_{\bar{X}} \), is equal to…
\( \sigma^2/n \)
\( n \) is the sample size. What is the value of \( \sigma^2_{\bar{X}} \) when \( n=1 \)?
The shape of \( \bar{X} \) depends on (possibly) two things: (1) the shape of \( X \) (2) the sample size \( n \).