using HypothesisTests

# Calculate the pairwise differences 
differences = after - before 

# Perform the paired t-test using the differences
t_test = OneSampleTTest(differences)

# Extract the p-value
p_value = pvalue(t_test)

# Print the results
println("Paired t-test Results:")
println("Test Statistic:", statistic(t_test))
println("p-value:", p_value)

# Interpretation
if p_value < 0.05
    println("Reject the null hypothesis. There is a statistically significant difference between the paired samples.")
else
    println("Fail to reject the null hypothesis. There is no statistically significant difference between the paired samples.")
end

Explanation:

1. Calculate the differences:

   • differences = after - before calculates the difference between each pair of measurements.

2. Perform the paired t-test:

   • OneSampleTTest(differences) performs a one-sample t-test on the calculated differences. This is equivalent to a paired t-test, as we are testing whether the mean difference between the pairs is significantly different from zero.

3. Extract the p-value: pvalue(t_test) extracts the p-value from the test result.

4. Print the results and interpretation: This part remains the same as in the previous example.

This approach directly utilizes the pairwise differences for the paired t-test, which is often considered more concise and mathematically accurate.