[SIX SIGMA GLOSSARY ALPHABETICAL INDEX] [SIX SIGMA GLOSSARY INDEX OF TOPICS]

• Hypothesis Test for the Mean
• One Sample t-Test
• Paired t-test
• t-test
• Two Sample t Test
• Two Sample z Test
• z Test

The hypothesis tests for the mean is a type of hypothesis test used to compare the means of two populations.

The tests in this topic are all parametric tests, and so require that the population conforms to a normal distribution. If the distribution is not normal then a nonparametric test should be used. Nonparametric tests for the mean include:

• Wilcoxon Signed Rank Test (nonparametric equivalent of the one sample t-test)
• Mann Whitney Test (nonparametric equivalent of the two sample t-test)

If the sample size is small (less than about thirty) and the process conforms to a normal distribution use the t-test. There are three choices:

• One Sample t-test
• Two sample t-test
• Paired t-test

For large samples, or where the process standard deviation is known and the process conforms to a normal distribution the choices are

• One sample z-test
• Paired z-test

The tests in this population compare only two populations. If there are more than two populations consider ANOVA.

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:

H₀ the population mean equals a specified value
H₁ the popular mean is [equal to/less than/greater than] a specified value

The test is:

where:

is the sample mean
m₀ is the specified value
s the sample standard deviation
n the sample size

The critical value of the t statistic t₀ can be found in t distribution tables, or the p-value 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 is 1 for a one sided test and 2 for a two sided test.

The t-test requires that the population conforms to a normal distribution.

The paired t-test is used when the test units can be paired. For example, the blood pressure of five patients was tested before and after a medication:

A Two Sample t test would show no significant difference because the difference between individuals masks the 'before' and 'after' difference.

The paired test involves carrying out a One Sample t-tests on the differences. Despite the variation between individuals the 'after' blood pressure is generally lower.

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:

H₀ the population means are equal
H₁ 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:

n₁, n₂ sample sizes

s₁, s₂ 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 where the standard deviation is known or where the sample size is large (greater than 30). The test requires that the population conforms to a normal distribution. The hypothesis is:

H₀ the population means are equal
H₁ the popular means are different

The test is:

The p-value can be obtained from Excel using the function:

one-tail test: = 1 - NORMSDIST(Z₀)
two-tail test: = (1 - NORMSDIST(Z₀))/2

Alternatively the critical values of the z statistic can be found from tables. For a one sided test:

A hypothesis test used to test the mean against a specified value. The test is used where the standard deviation is known or the sample is large (greater than about 30). The population must also conform to the normal distribution. The hypothesis is:

H₀ the population mean equals a specified value
H₁ the popular mean is [equal to/less than/greater than] a specified value

The test is:

where:

is the sample mean
m₀ is the specified value
s the sample standard deviation
n the sample size

The p-value can be obtained from Excel using the function:

one-tail test: = 1 - NORMSDIST(Z₀)
two-tail test: = (1 - NORMSDIST(Z₀))/2

Alternatively the critical values of the z statistic can be found from tables. For a one sided test: