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:nondirectional H0

H1:nondirectional H1


Directional:

H0:directional H0    or  directional H0

H1:directional H1    or  directional H1


 

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

 

formula one sample t test

 

whereas

 

estimated population standard deviation of the mean

 

and

 

estimated population standard deviation

 

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:

 

significance test 

* 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:

 

estimated population variance 

 

and we get the standard deviation of the mean:

 

estimated population variance of the mean 

 

now we can compute the t-value:

 

one sample t test significance test 

 

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.


Wiki link t-test




 

Gallery

 
 
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