Hypothesis Test for Paired Means

gossetThis test is sometimes called a Matched Pairs Hypothesis t Test. Matched pair design is when a person is tested against himself or very similar subjects matched in pairs. The randomness happens in which treatment is given first (if the subject is paired with himself) or which treatment is given to who (if contrasting the results of two subjects). This is NOT a two sample test. A two sample test compares the mean of an entire group to the mean of entirely different groups. Most importantly in two sample tests, the two groups being contrasted are independent. In a paired means test, we are looking at the difference for each pair, and they are dependent on one another. That is the point of the matched pairs test, to control for other variables that could affect our results.

The picture to the right is William Sealy Gosset, who under the pen name “Student” developed the t test.

There are 5 steps to a hypothesis Test.

Remember, we are performing this test on our new calculated variable, d. To perform a hypothesis test for paire means, apply the one-sample t procedures to the list of the differences for each pair.


List the what your null hypothesis and alternate hypothesis are in this step.

H_{0}: [ \mu_{d}=0]

Remember to include a full sentence describing what the null hypothesis means, in context. We are saying that the null hypothesis is that the mean value of the differences between our matched pairs are zero (or whatever value makes sense to use, in context. Zero is standard if we do not have another estimate already. This is saying that there is in fact NO difference between the two results for each pair).

H_{A}: [\mu_{d}\neq,<,>0]

Again remember to include a full sentence describing what the alternate hypothesis means, in context. Here, we are saying that the mean value of the differences between our matched pairs are different from the null hypothesis, so either greater than, less than, or not equal to whatever we defined in the null hypothesis.


1. Random? Each pair of measurements must be random. This site provides good information about randomization in matched pairs design (and also provides some sample questions to help you ensure you know when to use a paired samples test). Be sure to include any possible problems with the randomization.

2. 10% Condition? Our population must be larger than 10 times our sample. Be sure to include a discussion of what the population is and how we know our sample is an okay size.

3. Approximately normal? (this is for our new variable, d, the difference between the values for each pair). We will use the QQ plot and the histogram to show that the sample is nearly normal, unless n_{d} \ge40. By the central limit theorem, when n_{d} is greater than 40, our sample is approximately normal. This step insures that we can use a t distribution to reach a conclusion on the truth of the null hypothesis.


Remember degrees of freedom!! df=n_{d}-1


Where the numerator is the difference between the sample mean \bar{d} and the population mean (which comes from our null hypothesis, usually this will be a zero). The denominator is our standard error for our test statistic, and we find it by dividing the standard error by the square root of the sample size.

Also include a model of the distribution and shade where appropriate. Include the p-value that results from the test statistic you find using the formula.


For the conclusion, use the sentence we have for all hypothesis tests.

“Because my p-value of ____ is (greater than, less than) an alpha of 0.05 (or whatever you use as your standard), I (reject the null/fail to reject the null). The evidence does/does not seem to suggest that there is a difference between the two treatments (this part must be in context. For example, Because my p-value of 0.1986 is larger than an alpha of 0.05, I fail to reject the null. The evidence does not seem to suggest that a scented room helps students do better on tests.). Make sure you include the direction in which you subtracted. Because if there is in fact a difference, it matters in which way you subtracted to help the reader draw conclusions.

To complete the paired test in SPSS follow these steps:

Transform -> Compute Variable -> Target variable “d” -> subtract your variables in the compute field, and make sure to note which you are subtracting from the other.

Next, go to Analyze -> Descriptive Statistics -> explore and select d as the dependent list.

also select the plots option ->and check histogram and normality plot.

Next Analyze -> Compare means -> Paired samples t test.

Choose variables based on the order you want to subtract (this can help your interpretation of the results of the test if you subtract in a certain order).

Run the test.

This site provides a nice example of a matched pairs design. It can give you a concrete idea of how a matched pairs test would be used in real life and it also includes some good commentary on randomization when conducting a matched pairs experiment.

This site goes very in-depth and explains all aspects of the paired means t test. If you are confused about any aspect of this test, this webpage will probably have the answer. It includes examples and a thorough explanation of everything pertaining to the hypothesis test.


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