Sampling Distribution of the Sample Mean

PROBABILITY AND STATISTICS
Lab, Seminar, Lecture 4.

Behavior of the sample average

X-bar
 The topic of 4th seminar&lab is the average of the

population that has a certain characteristic.  This average is the population parameter of interest, denoted by the greek letter mu.  We estimate this parameter with the statistic x-bar, the average in the sample.

Probability and statistics - Karol Flisikowski

X-bar Definition

1 x   xi n i 1
Probability and statistics - Karol Flisikowski

n

Sampling Distribution of x-bar
 How does x-bar behave? To study the behavior,

imagine taking many random samples of size n, and computing an x-bar for each of the samples.  Then we plot this set of x-bars with a histogram.

Probability and statistics - Karol Flisikowski

Sampling Distribution of x-bar

Probability and statistics - Karol Flisikowski

Central Limit Theorem
 The key to the behavior of x-bar is the central limit

theorem. It says:  Suppose the population has mean, m, and standard deviation s.  Then, if the sample size, n, is large enough, the distribution of the sample mean, x-bar will have a normal shape, the center will be the mean of the original population, m, and the standard deviation of the x-bars will be s divided by the square root of n.
Probability and statistics - Karol Flisikowski

Central Limit Theorem
 If the CLT holds we have,

 Normal shape
 Center = mu

 Spread = sigma/sqroot n.

Probability and statistics - Karol Flisikowski

When Does CLT Hold?
 Answer generally depends on the sample size, n,

and the shape of the original distribution.  General Rule: the more skewed the population distribution of the data, the larger sample size is needed for the CLT to hold.

Probability and statistics - Karol Flisikowski

CLT

 Previous overhead shows the original population distribution in (a),

and increasing sample sizes through graphs (b), (c), and (d).  Notice that it takes