T Test Probability

By Author: Pierce Brosnan
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Introduction to t test probability:

T TEST :

A continuous probability distribution which has the following probability density function is called student’s t test probability distribution.

f(t) = `1/(sqrt(n)beta(1/2,n/2))` `1/(1 + t^2/n)^((n+1)/2)`

-`oo` < t < `oo`

Here the variable t is called student’s t test - variable with n-degree of freedom.

Note

1. t-test probability distribution has one parameter, namely n which is called degree of freedom.

2. t-test probability distribution has range (-8, 8) and it is symmetrical.

3. t-test probability variate with n degrees of freedom is denoted by t(n).

4. ß(1/2,n/2) is a positive constant.

5.This distribution was introduced by W.S.Gosset who published articles in the name Student.

Properties of Student’s T Test Probability Distribution

1. t test probability distribution's range is (-8, 8)

2.t test probability distribution is symmetrical and bell shaped

3.t test probability distribution's mean =0

4.t test probability distribution's variance is n/(n-2) for n> ...
... 2 and is greater than 1.

5.t test probability distribution is leptokurtic. 6. For arge n, student’s t test probability distribution variate tends to Standard Normal Variate.

Applications of T Test Probability Distribution

The situations where t test probability distribution distribution is applicable are

1. to test whether a population has a given mean.

2. to test whether the means of two populations are equal.

3. to test whether the means of two populations are equal when the observations are paired.

Example Problems on T Test Probability Distribution:

Discuss the factors that affect the magnitude of the t statistic and whether you will reject your null hypothesis. Make sure to describe how each one of these affects the t statistic

[Use t test probability distribution]

Answer of T Test Probability Distribution:

Factors that effect the magnitude of the t test probability distribution are;

•The null hypothesis H0 is not rejected unless there is strong evidence against it.

•If one of the two hypotheses is 'simpler' then it is given priority so that a more 'complicated' theory is not adopted unless there is sufficient evidence against the simpler one ( "If there are two possible explanations always accept the simplest")

•In general, it is 'simpler' to propose: there is no difference between two sets of results than to say that there is a difference.

The outcome of a hypothesis testing can be "reject H0" or "do not reject H0". If we say that "do not reject H0", this does not always mean that the null hypothesis is true, only that there is insufficient evidence against H0 in favour of HA. Rejecting the null hypothesis mean that the alternative hypothesis may be true.

In order to decide whether to accept or reject the null hypothesis, the level of significance (`alpha` ) of the result is used `alpha = 0.05 or alpha= 0.01`. This allows us to state whether or not there is a "significant difference” between populations.

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