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A paired t-test by hand is a statistical test used to compare two sets of data. It is done by subtracting the two sets of data, then calculating the mean and standard deviation of the differences between the two sets. The t-statistic is then calculated by taking the difference in means, divided by the standard error of the difference. The final step is to compare the t-statistic to the critical t-value in the t-table to determine the probability of the results.
A is used to compare the means of two samples when each in one sample can be paired with an observation in the other sample.
The following step-by-step example shows how to perform a paired samples t-test to determine if the population means are equal between the following two groups:
Step 1: Calculate the Test Statistic
The test statistic of a paired t-test is calculated as:
t = xdiff / (sdiff/√n)
where:
- xdiff: sample mean of the differences
- s: sample standard deviation of the differences
- n: sample size (i.e. number of pairs)
We will calculate the mean of the differences between the two groups and the standard deviation of the differences between the two groups:
Thus, our test statistic can be calculated as:
- t = xdiff / (sdiff/√n)
- t = 1.75 / (1.422/√12)
- t = 4.26
Step 2: Calculate the Critical Value
Next, we need to find the critical value to compare our test statistic to.
For this example, we’ll use a two-tailed test with α = .05 and df = n-1 degrees of freedom.
According to the , the critical value that corresponds to these values is 2.201:
Step 3: Reject or Fail to Reject the Null Hypothesis
Our paired samples t-test uses the following null and alternative hypothesis:
- H0: μ1 = μ2 (the two population means are equal)
- HA: μ1 ≠ μ2 (the two population means are not equal)
Since the absolute value of our test statistic (4.26) is greater than the critical value found in the t-table (2.201), we reject the null hypothesis.
This means we have sufficient evidence to say that the mean between the two groups is not equal.
Bonus: Feel free to use the to confirm your results.