**Definition**: Used to compare
means on the same or related subject over time or in differing circumstances.

**Assumptions:** The observed
data are from the same subject or from a matched subject and are drawn from a
population with a normal distribution.

**Characteristics:** Subjects
are often tested in a before-after situation (across time, with some
intervention occurring such as a diet), or subjects are paired such as with
twins, or with subject as alike as possible. An extension of this test is the
repeated measure ANOVA.

**Test:** The paired t-test is
actually a test that the differences between the two observations is 0. So, if
*D* represents the difference between observations, the hypotheses are:

H_{o}:* D* = 0 (the
difference between the two observations is 0)

H_{a}: *D *0 (the
difference is not 0)

The test statistic is *t* with
n-1 degrees of freedom. If the p-value associated with t is low (< 0.05), there
is evidence to reject the null hypothesis. Thus, you would have evidence that
there is a difference in means across the paired observations.

**Location in WINKS:** The
paired t-test is located in the "Analysis --> t-test and Analysis of Variance" menu.

**See also:** Repeated Measures
Analysis of Variance. Also, if the differences have already been calculated, a
single sample test of u = 0 would be equivalent to the paired t-test. The
non-parametric counterpart to the paired t-test is Friedman's test.

Example in WINKS: Paired t-test

The DIET.DBF database contains
information on 8 subjects who were placed on a diet, and observed for several
weeks. Before and after weights were taken.

**Step 1: **In WINKS, oOpen the data set named** DIET.SDA** (Or create a data set with the data.)

**Step 2**: Select **Analyze, t-tests and ANOVA/Paired/Rep. Measure (t-test/ANOVA).**

**Step 3:** Select *REP1* (*BEFORE*) as the first field and *REP2(AFTER)* as the second field and **click Ok**.

**Step 4:** The results are shown in the viewer. A portion of the output is shown below:

-----------------------------------------------------------------
Repeated Measures Analysis Summary C:\WINKS\DIET.DBF
-----------------------------------------------------------------
Number of repeated measures is 2
Number of subjects read in 8

-----------------------------------------------------------------
Means and standard deviations for 2 repeated measures:
-----------------------------------------------------------------

1)REP1: mean = 169.625 s.d. = 8.07001
2)REP2: mean = 150.25 s.d. = 11.04213

Mean Difference = 19.375 s.d.(difference) = 14.78356

95% C.I. about Mean Difference is (7.01367, 31.73633)

Paired t- test

Calculated t = 3.70687 with 7 D.F. p = 0.0076 (two- sided)

The means and standard deviations for each group are reported, but more importantly, the mean difference between BEFORE and AFTER measurements is given. The statistical procedures are performed on this average difference. These results are interpreted like those of a single sample t-test with null hypothesis: mean=0, and alternative hypothesis: mean <> 0.

The calculated t-statistic is t = 3.70687. The test is performed with 7 degrees of freedom, and p = 0.0076. A small p-value such as this is indicates rejection of the null hypothesis and leads to the conclusion that the average difference in BEFORE and AFTER weights is not zero, i.e., there is evidence of a significant (at the 0.05 level) change of weight in these eight subjects on average.

**Step 5:** Select **Graph, Display Graph** to display a graphical comparison.

**More about WINKS** >>

Exercise - Paired t-test

A company was wondering which style
of pepperoni pizza was most popular. It set up an experiment where ten people
were each given two types of pizza to eat, Type A and Type B. Each pizza was
carefully weighed at exactly 16 oz. After fifteen minutes, the remainders of the
pizza were weighed, and the amount of each type pizza remaining per person was
calculated. It is assumed that the subject would eat more of the type of pizza
he or she preferred. Here are the data:

Subject |
Pizza A |
Pizza B |

1 |
12.9oz |
16oz |

2 |
5.7 |
7.5 |

3 |
16 |
16 |

4 |
14.3 |
15.7 |

5 |
2.4 |
13.2 |

6 |
1.6 |
5.4 |

7 |
14.6 |
15.5 |

8 |
10.2 |
11.3 |

9 |
4.3 |
15.4 |

10 |
6.6 |
10.6 |

1. Perform a paired t-test.

2. What value of the t-statistic is
calculated, what are the degrees of freedom for the test, what p-value is
reported for this data?

3. Do people seem to prefer one
type of pepperoni pizza over another? If so, which seems to be most liked?

**Do this analysis right NOW: WINKS SDA Statistical Software & Graphs** provides extensive statistics in a professional and understandable software program designed from the ground up to be easy to learn and use.