B.Pharmacy

The paired t-test is a statistical method used to compare the means of two related groups to determine whether there is a significant difference between them. It's typically applied in situations where the same subjects are measured twice, such as before and after an intervention, or when two related but distinct measures are taken from the same group. This test assumes that the data are normally distributed.

Key Concepts:

Paired data: The data sets are "paired" because each subject in one group has a corresponding subject in the other group. For example, measuring the same people before and after a treatment.
Null Hypothesis (H₀): The null hypothesis assumes that there is no difference between the two paired means (i.e., the mean difference is zero).
Alternative Hypothesis (H₁): The alternative hypothesis assumes that there is a significant difference between the two paired means (i.e., the mean difference is not zero).

Formula:

The formula for the paired t-test is:

t = \frac{\overline{d}}{s_{d} / \sqrt{n}}

Where:

$\overline{d}$ = the mean of the differences between the paired observations.
$s_{d}$ = the standard deviation of the differences.
$n$ = the number of pairs.

Steps to Perform a Paired T-Test:

Collect Paired Data: Obtain two sets of measurements for each individual or subject.
Calculate the Difference: For each pair of measurements, subtract one value from the other to get the difference.
Calculate the Mean and Standard Deviation of the Differences: Find the mean ( $\overline{d}$ ) and standard deviation ( $s_{d}$ ) of the differences.
Calculate the t-statistic: Use the formula above to calculate the t-statistic.
Determine the Degrees of Freedom (df): The degrees of freedom for a paired t-test is $n - 1$ , where $n$ is the number of pairs.
Compare to Critical Value: Compare the calculated t-statistic to the critical value from the t-distribution table based on the degrees of freedom and desired significance level (usually 0.05). Alternatively, calculate the p-value and compare it to the significance level.
Make a Decision: If the t-statistic is greater than the critical value or if the p-value is smaller than the significance level, reject the null hypothesis, indicating a significant difference between the paired groups.

Assumptions:

Normality: The differences between pairs should be approximately normally distributed.
Independence: Each pair of observations should be independent of other pairs.
Scale of Measurement: The data should be measured on a continuous scale (interval or ratio).

Example:

Suppose a group of patients is given a drug, and their blood pressure is measured before and after the treatment. A paired t-test could be used to test whether there is a significant change in blood pressure due to the drug.

Before: [120, 130, 140, 150, 160]
After: [115, 128, 138, 145, 158]

By calculating the differences and running the t-test, you can determine if the observed changes are statistically significant or likely due to chance.

The paired t-test is a powerful tool for analyzing changes within subjects or related groups over time or under different conditions.