Two-Sample t Test

About the Two-Sample t Test Task

A two-sample t test compares the mean of the first sample minus the mean of the second sample to a given number, the null hypothesis difference.

To compare means from two independent samples with n₁ and n₂ observations to a value m, use $t equals . fraction open , modified x sub 1 with macron above , minus , x with macron above sub 2 , close minus m , over s . square root of fraction 1 , over n sub 1 end fraction , plus , fraction 1 , over n sub 2 end fraction , end root end fraction. Click image for alternative formats.$ . In this example, s² is the pooled variance $s squared , equals . fraction open , n sub 1 , minus 1 close , s sub 1 and super 2 , plus open , n sub 1 , minus 1 close , s sub 2 and super 2 , , over n sub 1 , plus , n sub 2 , minus 2 end fraction. Click image for alternative formats.$ , and s²₁ and s²₂ are the sample variances of the two groups. The use of this t statistic depends on the assumption that

sigma sub 1 and super 2 , equals , sigma sub 2 and super 2. Click image for alternative formats.

, where

sigma sub 1 and super 2. Click image for alternative formats.

and

sigma sub 2 and super 2. Click image for alternative formats.

are the population variances of the two groups.

To run a two-sample t test, open the t Tests task. From the t test drop-down list, select Two-sample test.

Example: Two-Sample t Test

In this example, you want to analyze the height values for males and females in your class.

To create this example:

In the Tasks section, expand the Statistics folder, and then double-click t Tests. The user interface for the t Tests task opens.
On the Data tab, select the SASHELP.CLASS data set.

Tip
If the data set is not available from the drop-down list, click . In the Choose a Table window, expand the library that contains the data set that you want to use. Select the data set for the example and click OK. The selected data set should now appear in the drop-down list.
From the t test drop-down list, select Two-sample test.

Assign columns to these roles:

Role	Column Name
Analysis variable	Height
Groups variable	Sex

To run the task, click .

Here is a subset of the results:

Distribution of Height for Males and Females

Assigning Data to Roles

To run a two-sample t test, you must select an input data source. To filter the input data source, click Filter Icon

Next, select Two-sample test from the t test drop-down list. Assign a column to the Analysis variable and Groups variable roles.

Setting Options

Option Name	Description
Tests
Tails	specifies the number of sides (or tails) and direction of the statistical tests and test-based confidence intervals. You can choose from these options: Two-tailed test specifies two-sided tests and confidence intervals for means. Upper one-tailed test specifies upper one-sided tests in which the alternative hypothesis indicates a mean greater than the null value, and upper one-sided confidence intervals between the lower confidence limit and infinity. Lower one-tailed test specifies lower one-sided tests in which the alternative hypothesis indicates a mean less than the null value, and lower one-sided confidence intervals between minus infinity and the upper confidence limit.
Alternative hypothesis	specifies the value of the null hypothesis.
Cox and Cochran probability approximation for unequal variances	calculates the Cochran and Cox approximation. This approximation of the p-value of the t_u is the value of p such that $t sub u , equals . fraction open . fraction s sub 1 and super 2 , over sum , from , i equals 1 , to , n with subscript 1 , and with superscript times , end sub-superscript , of . f sub 1 i end sub . w sub 1 i end sub end fraction . close , t sub 1 , plus open . fraction s sub 2 and super 2 , over sum , from , i equals 1 , to , n with subscript 2 , and with superscript times , end sub-superscript , of . f sub 2 i end sub . w sub 2 i end sub end fraction . close , t sub 2 , over open . fraction s sub 1 and super 2 , over sum , from , i equals 1 , to , n with subscript 1 , and with superscript times , end sub-superscript , of . f sub 1 i end sub . w sub 1 i end sub end fraction . close plus open . fraction s sub 2 and super 2 , over sum , from , i equals 1 , to , n with subscript 2 , and with superscript times , end sub-superscript , of . f sub 2 i end sub . w sub 2 i end sub end fraction . close end fraction. Click image for alternative formats.$ . In this example, t₁ and t₂ are the critical values of the t distribution corresponding to a significance level of p and sample sizes n₁ and n₂, respectively. The degrees of freedom is undefined when . (Cochran and Cox 1950).
Normality Assumption
Tests for normality	runs tests for normality that include a series of goodness-of-fit tests based on the empirical distribution function. The table provides test statistics and p-values for the Shapiro-Wilk test (provided the sample size is less than or equal to 2000), the Kolmogorov-Smirnov test, the Anderson-Darling test, and the Cramér-von Mises test.
Nonparametric Tests Note: This option is available only for a two-tailed test when the alternative hypothesis equals 0.
Wilcoxon rank-sum test	generates an analysis of Wilcoxon scores. When there are two classification levels (samples), this option produces the Wilcoxon rank-sum test.
Plots
Histogram and box plot	creates a histogram and box plot together in a single panel, sharing common X axes.
Normality plot	creates a normal quantile-quantile (Q-Q) plot.
Confidence interval plot	creates plots of the confidence interval for means. This plot is not created by default.
Wilcoxon box plot	creates a box plot of Wilcoxon scores. This plot is associated with the Wilcoxon analysis. This plot is not created by default. Note: This plot is available only for a two-tailed test when the alternative hypothesis equals 0.