Methods and formulas for 2-Sample t

Select the method or formula of your choice.

Confidence interval (CI)

Term	Description
	mean of the first sample
	mean of the second sample
t_α_/2	inverse cumulative probability of a t distribution at 1 – α/2
α	1 - confidence level / 100
s	sample standard deviation as calculated for the test statistic

The sample standard deviation, s, of

depends upon the variance assumption.

Unequal variances

When you assume unequal variances, the sample standard deviation of is:

The degrees of freedom are:

If necessary, Minitab truncates the degrees of freedom to an integer, which is a more conservative approach than rounding.

Equal variances

When you assume equal variances, the common variance is estimated by the pooled variance:

The standard deviation of

is estimated by:

The test statistic degrees of freedom are:

DF = n₁ + n₂ – 2

Term	Description
	mean of the first sample
	mean of the second sample
s	sample standard deviation of
δ₀	hypothesized difference between the two population means
s₁	sample standard deviation of the first sample
s₂	sample standard deviation of the second sample
n₁	sample size of the first sample
n₂	sample size of the second sample
VAR₁
VAR₂

Suppose C1 contains the response, and C3 contains the mean for each factor level. For example:

Choose Calc > Calculator.
In Store result in variable, enter C4.
In Expression, enter SQRT((SUM((C1 - C3)**2)) / (total number of observations - number of groups)) . For the previous example, the Expression for the pooled standard deviation would be: SQRT((SUM(('Response' - 'Mean')**2)) / (6 - 2))

The value that Minitab stores is 3.75489.

The calculation for the p-value depends on the alternative hypothesis.

The degrees of freedom, DF, depend upon the variance assumption.

Unequal variances

When you assume unequal variances, the degrees of freedom are:

If necessary, Minitab truncates the degrees of freedom to an integer, which is a more conservative approach than rounding.

Equal variances

When you assume equal variances, the test statistic degrees of freedom are:

DF = n₁ + n₂ – 2

Term	Description
μ₁	population mean of the first sample
μ₁	population mean of the second sample
n₁	sample size of the first sample
n₂	sample size of the second sample
δ₀	hypothesized difference between the two population means
t	t-statistic from the sample data
t	a random variable from the t-distribution with DF degrees of freedom.
VAR₁
VAR₂