Statistics
Tutorial
This is one in a series of tutorials using examples from WINKS SDA Software
Paired t-test
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:
Ho: D = 0 (the difference between the two observations is 0)
Ha: 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 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: 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. The results of a paired t-test analysis are:
----------------------------------------------------------------- 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 t-test reports t = 3.71 and p = .008. For this evidence, you can conclude that the diet produced a significant amount of weight loss (look at the before and after means.)
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?
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