A paired sample t-test is used to determine whether there is a significant difference between the average values of the same measurement made under two different conditions. Both measurements are made on each unit in a sample, and the test is based on the difference between these two values. The usual null hypothesis is “the difference in the mean values is zero”. For example, the yield of two strains of barley is measured in successive years in twenty different plots of agricultural land (the units) to investigate whether one crop gives a significantly greater yield than the other, on average (1).

## When to use T-test?

You use a paired-samples (also known as the matched cases) T test if you want to test whether 2 population means are equal, and you have 2 measurements from pairs of people or objects that are similar in some important way. In other words, you are saying that the difference between the mean values of two given sets of samples is 0. Some examples wherein Paired Sample T-Test will be applied are given below:

## Examples

- You are interested in determining whether self-reported weights and actual weights differ. You ask a random sample of 200 people how much they weigh and then you weigh them on a scale. You want to compare the means of the 2 related sets of weights.
- You want to test the null hypothesis that husbands and wives have the same average years of education. You take a random sample of married couples and compare their average years of education.
- You want to compare 2 methods for teaching reading. You take a random sample of 50 pairs of twins and assign each member of a pair to one of the 2 methods. You compare average reading scores after completion of the program.

## Sample analysis

In a paired-samples design, both members of a pair must be on the same data record. Different variable names are used to distinguish the 2 members of a pair. For example: A physician is evaluating a new diet for her patients with a family history of heart disease. To test the effectiveness of this diet, 16 patients are placed on the diet for 6 months. Their weights and triglyceride levels are measured before and after the study, and the physician wants to know if either set of measurements has changed.

**Null hypothesis:** There is no difference in the levels of Triglycerides and weight of individual after using new diet for 6 months.

**Null hypothesis:** There is has been a significant difference in the levels of Triglycerides and weight of individual after using new diet for 6 months.

We are using Paired-Samples T Test to determine whether there is a statistically significant difference between the pre- and post-diet weights and triglyceride levels of these patients.

**Step 1:**Select “Analyze”, then “Compare means”, then “Paired-samples T test” (See Figure 1)

**Step 2:**Select “Triglyceride” and “Final Triglyceride” as the first set of paired variables.

**Step 3:**Select “Weight” and “Final weight” as the second pair

**Step 4:**After selecting the pairs, click on “Options” and set the Confidence Interval Percentage at 99% (by default it is 95%, you can change it as per your requirement).**Step 5:**Lastly click “OK”.

**Legends (Table 1):**

- The
**Mean**column in the paired-samples t test table displays the average difference between triglyceride and weight measurements before the diet and six months into the diet. - The
**Std. Deviation**column displays the standard deviation of the average difference score. - The
**Std. Error Mean**column provides an index of the variability one can expect in repeated random samples of 16 patients similar to the ones in this study. - The 95% Confidence Interval of the Difference provides an estimate of the boundaries between which the true mean difference lies in 95% of all possible random samples of 16 patients similar to the ones participating in this study.
- The
**t statistic**is obtained by dividing the mean difference by its standard error. - The
**Sig. (**2-tailed) column displays the probability of obtaining a t statistic whose absolute value is equal to or greater than the obtained t statistic

The Descriptives table displays the mean, standard deviation, and standard error for both groups. The information is disseminated in pairs such that pair 1 should come first and pair 2 should come second in the table.

The “Mean” section denotes the difference between averages of the two samples for each pair. Therefore, in case of Pair 1 (Triglyceride & Final Triglyceride), the difference in means of the two sample sets (triglyceride levels after the diet period of 6 months) is 14.06.

The subjects clearly lost weight over the course of the study; on average, about 8 pounds.

The standard deviations for pre- and post-diet measurements reveal that subjects were more variable with respect to triglyceride levels (46.875) than weight (2.886).

At 0.249, the Pearson correlation between the baseline and six-month triglyceride levels is not statistically significant. Levels were lower overall, but the change was inconsistent across subjects. Several lowered their levels, but several others either did not change or increased their levels.

On the other hand, the Pearson correlation between the baseline and six-month weight measurements is 0.000, almost a perfect correlation. Unlike the triglyceride levels, all subjects lost weight and did so quite consistently.

Conclusions which can be drawn: Since the significance value for change in weight is less than 0.01, you can conclude that the average loss of 8.06 pounds per patient is not due to chance variation, and can be attributed to the diet.

However, the significance value greater than 0.10 for change in triglyceride level shows the diet did not significantly reduce their triglyceride levels.

**References**

- Paired Sample t-test | the Statistics Glossary. (n.d.). The Statistics Glossary. Retrieved August 07, 2014, from http://www.stats.gla.ac.uk/glossary/?q=node/355.

### Priya Chetty

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