CK-12 Probability and Statistics - Advanced

http://www.ck12.org Chapter 7. Sampling Distributions and Estimations

Since thet-distribution is symmetric about a mean of zero, the following statement is true.

tα=−t 1 −α and t 1 −α=−tα

Therefore, ift 0. 05 = 2 .92 then by applying the above statementt 0. 95 =− 2. 92

At-distribution is mound shaped, with mean 0 and a spread that depends on the degrees of freedom. The greater the

degrees of freedom, the smaller the spread. As the number of degrees of freedom increases, thet-distribution ap-

proaches a normal distribution. The spread of anyt-distribution is greater than that of a standard normal distribution.

This is due to the fact that that in the denominator of the formulaσhas been replaced withs. Sincesis a random

quantity changing with various samples, the variability intis greater, resulting in a larger spread.

Notice in the first distribution graph the spread of the first (inner curve) is small but in the second one the both

distributions are basically overlapping, so are roughly normal. This is due to the increase in the degrees of freedom.

Here are thet-distributions ford f=1 and ford f=12 as graphed on the graphing calculator

You are now on theY=screen.

Y=tpdf(X, 1 )[Graph]

Repeat the steps to plot more than onet-distribution on the same screen.

Notice the difference in the two distributions.

The one with 12=d fapproximates a normal curve.

Thet-distribution can be used with any statistic having a bell-shaped distribution. The Central Limit Theorem states

the sampling distribution of a statistic will be close to normal with a large enough sample size. As a rough estimate,

the Central Limit Theorem predicts a roughly normal distribution under the following conditions:

1. The population distribution is normal.


2. The sampling distribution is symmetric and the sample size is≤15.