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Hypothesis Tests |

The Paired *t*-test enables you to determine whether
the means of paired samples are equal.
The term *paired*
means that there is a correspondence between observations from each
population. For example, the birth and death data analyzed in the
preceding section are considered to be paired data because, in each
observation, the variables birth and death correspond to
the same state.

Suppose that you want to determine whether the means for the birth
rate and the infant mortality rate are equal. Analyst provides the
Two-Sample Paired *t*-test for Means task, which tests the equality of
means of two
paired samples. The two samples in this example are the birth rate
(birth) and the infant mortality rate (death) for each state.

- Select
**Tools****Sample Data**... - Select Bthdth92.
- Click
**OK**to create the sample data set in your Sasuser directory. - Select
**File****Open By SAS Name**... - Select Sasuser from the list of
**Libraries**. - Select Bthdth92 from the list of members.
- Click
**OK**to bring the Bthdth92 data set into the data table.

- Select
**Statistics****Hypothesis Tests****Two-Sample Paired t-test for Means**... - Select the variable birth as the Group 1 variable.
- Select the variable death as the Group 2 variable.

The test of interest is whether the difference of the means is zero. This is the default value in Analyst, although you can specify other values as well.

You can choose one of three alternative hypotheses. The default is that the difference between the means is not equal to the specified difference, which is the two-sided alternative. The one-sided alternatives are that the difference is greater than, or less than, the difference specified in the null hypothesis.

In Figure 8.7, the null hypothesis specifies that the means of the variables birth and death are equal (or, equivalently, that the difference between the means is 0). The alternative hypothesis is that the two means are not equal.

- Click on the
**Plots**button in the main dialog. - Select
**Box-&-whisker plot**. - Select
**Means plot**. - Click
**OK**.

Figure 8.8 displays the Plots dialog
with **Box-&-whisker plot** and **Means plot** selected.

Click **OK** in the main dialog
to perform the analysis.

In Figure 8.9, the "Sample Statistics" table shows that the
mean of the variable birth is larger than that of the
variable death. In the "Hypothesis Test" table, the *
t* statistic (19.926) and associated *p*-value (< 0.0001)
indicate that the difference between the two means is
statistically very significant.

Figure 8.10 displays the side-by-side box plots of birth and death. Observations that fall beyond the whiskers are individually identified with a square symbol.

The means and standard error plot displayed in Figure 8.11 provides another view of the two variables. The means plot depicts an interval centered on the sample mean for each variable. The vertical line interval extends two standard deviations on either side of the mean.

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