# One-sample t-test

## One sample t-test

Comparison of sample mean with population mean when standard deviation of the population is estimated from the sample.

## Requirements:

Dependent variable is at least interval scaled and stems from a normal distributed population.

## Hypothesis:

Nondirectional:

H0:

H1:

Directional:

H0:    or

H1:    or

The term   is t-distributed with n-1 degrees of freedom.

whereas

and

The obtained t-value is then compared with the critical t-value to make the decision of significance.

However, for sufficient large samples (n > 100)* one can compute the z-Value even if σ is estimated from the sample:

* for large samples (n > 100) the t-distribution approximates the standard normal distribution sufficiently well.

## Example of a one sample t-test

A random sample of 22 fifth grade pupils have a grade point average of 5.0 in maths with a standard deviation of 0.452, whereas marks range from 1 (worst) to 6 (excellent). The grade point average (GPA) of all fifth grade pupils of the last five years is 4.7. Is the GPA of the 22 pupils different from the populations’ GPA?

 Pupil Grade points 1 5 2 5.5 3 4.5 4 5 5 5 6 6 7 5 8 5 9 4.5 10 5 11 5 12 4.5 13 4.5 14 5.5 15 4 16 5 17 5 18 5.5 19 4.5 20 5.5 21 5 22 5.5 Mean 5.0 Variance 0.2045

We estimate the standard deviation of the population:

and we get the standard deviation of the mean:

now we can compute the t-value:

for the non directed Hypothesis we have a critical t-value of t(df=21, alpha=0.05)=2.080

for the directed Hypothesis we have a critical t-value of t(df=21, alpha=0.05)=1,721

The obtained t-value is greater that the critical t-value. The grade point average of the 22 pupils is different from the populations’ GPA.

BrightStat output of one sample t-test example

This is a fictitious example.

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