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Gather Your Inputs
First, identify the paired measurements. Let's denote the first measurement as X1 and the second measurement as X2. For example, X1 could be the weight of subjects before a diet, and X2 could be the weight of the same subjects after the diet.
Calculate the Differences
Next, calculate the differences between the paired measurements. Let's denote the differences as d = X2 - X1. For example, if the weights before the diet are 70, 65, 80, and 75 kg, and the weights after the diet are 65, 60, 75, and 70 kg, the differences would be -5, -5, -5, and -5 kg.
Calculate the Mean and Standard Deviation of the Differences
Then, calculate the mean (d̄) and standard deviation (s) of the differences. The mean is calculated as the sum of the differences divided by the number of pairs. The standard deviation is calculated as the square root of the sum of the squared differences divided by the number of pairs minus 1.
Apply the Paired t-Test Formula
The paired t-test formula is t = d̄ / (s / √n), where d̄ is the mean of the differences, s is the standard deviation of the differences, and n is the number of pairs. For example, if the mean of the differences is -5 kg, the standard deviation of the differences is 0 kg, and the number of pairs is 4, the t-value would be -5 / (0 / √4) = -5 / 0 = undefined, indicating that the standard deviation is zero, and the formula cannot be applied. In this case, the paired t-test is not suitable, and an alternative test should be used.
Determine the Degrees of Freedom and Look Up the Critical t-Value
The degrees of freedom for the paired t-test is n-1, where n is the number of pairs. Look up the critical t-value in a t-distribution table using the degrees of freedom and the chosen significance level (usually 0.05). For example, if the number of pairs is 4, the degrees of freedom would be 3, and the critical t-value for a two-tailed test at a significance level of 0.05 would be approximately 3.182.
Interpret the Results
Finally, compare the calculated t-value to the critical t-value. If the calculated t-value is greater than the critical t-value, the null hypothesis is rejected, indicating that there is a significant difference between the paired measurements. For example, if the calculated t-value is -5 / (0 / √4) is undefined, and an alternative test should be used. However, if the calculated t-value is 4 and the critical t-value is 3.182, the null hypothesis is rejected, indicating that there is a significant difference between the paired measurements.
Introduction to Paired t-Test
The paired t-test is a statistical test used to compare the means of related or paired measurements. This test is commonly used in experiments where the same subjects are measured before and after a treatment, or where subjects are matched in pairs.
When to Use Paired t-Test
The paired t-test is used when the measurements are related, such as:
- Before and after a treatment
- Matched pairs of subjects
- Repeated measurements on the same subject
Step-by-Step Calculation
To perform a paired t-test calculation by hand, follow these steps: