Hypothesis Test for Means


For small samples (less than about thirty) and the process conforms to a normal distribution use the ttest. Choices are:
One Sample ttest
Two sample ttest
Paired ttest
For large samples, or where the process standard deviation is known and the process conforms to a normal distribution the choices are
One sample ztest
Paired ztest
Nonparametric tests for the mean include:
Wilcoxon Signed Rank Test (nonparametric equivalent of the one sample ttest)
Mann Whitney Test (nonparametric equivalent of the two sample ttest)
Use ANOVA for more than two samples.
A hypothesis test used to test the mean of a small sample taken from a population with a normal distribution against a specified value. The hypothesis:
H_{0} the population mean equals a specified value
H_{1} the popular mean is [equal to/less than/greater than] a specified value
The test is:
where:
is the sample mean
m_{0} is the specified value
s the sample standard deviation
n the sample size
The critical value of the t statistic t_{0} can be found in t distribution tables, or the pvalue can be found using the Excel function:
=TDIST(t0, n, Tails)
The number of degrees of freedom 'n' is n  1, the number of tails depends on whether it is a one or two sided test.
The ttest requires that the population conforms to a normal distribution.
The paired ttest is used when the test units can be paired. For example, the blood pressure of five patients was tested before and after a medication:
Patient 
Before

After

D

Andrew 
120

110

10

Bill 
135

115

20

Charles 
110

110

0

David 
140

135

5

Eric 
115

110

5

A Two Sample t test would show no significant difference because the standard deviations of the 'before' and 'after' columns is high.
The paired test involves carrying out a One Sample ttests on the differences.
See Hypothesis Test for Means
A hypothesis test used to to compare the means of two reasonably small (30 or less) samples to see if it is feasible that they come from the same population. The hypothesis is:
H_{0} the population means are equal
H_{1} the popular means are different
The test is:
Step 1: calculate the pooled standard deviation
Step 2: calculate the t statistics
The degrees of freedom:
n_{1}, n_{2} sample sizes
s_{1}, s_{2} sample standard deviation
A hypothesis test used to to compare the means of two samples to see if it is feasible that they come from the same population. The test is used with large samples (greater than 30) taken from a normal distribution. The hypothesis is:
H_{0} the population means are equal
H_{1} the popular means are different
The test is:
The pvalue can be obtained from Excel using the function:
onetail test: = 1  NORMSDIST(Z_{0})
twotail test: = (1  NORMSDIST(Z_{0}))/2
Alternatively the critical values of the z statistic can be found from tables. For a one sided test:
a 
0.10

0.05

0.025

0.01

Z_{a} 
1.28

1.64

1.96

2.33

A hypothesis test used to test the mean of a large sample (greater than about 30) against a specified value. The hypothesis:
H_{0} the population mean equals a specified value
H_{1} the popular mean is [equal to/less than/greater than] a specified value
The test is:
where:
is the sample mean
m_{0} is the specified value
s the sample standard deviation
n the sample size
The pvalue can be obtained from Excel using the function:
onetail test: = 1  NORMSDIST(Z_{0})
twotail test: = (1  NORMSDIST(Z_{0}))/2
Alternatively the critical values of the z statistic can be found from tables. For a one sided test:
a 
0.10

0.05

0.025

0.01

Z_{a} 
1.28

1.64

1.96

2.33

