Paired T-Distribution, Paired T-Test, Paired Comparison Test, Paired Sample Test

A Paired T-Distribution and Paired T-Test (Paired T-Distribution, Paired T-Test, Paired Comparison Test, Paired Sample Test) are statistical methods that compare the mean and standard deviation of two matched groups to determine if there is a significant difference between the two groups. In other words, they test whether or not the average difference between the two measurements is statistically significant from zero.

Student’s T-Distribution is used to find confidence intervals for the population mean when the sample size is less than 30 and the population standard deviation is unknown.  Further, the Student’s T-Test is divided into paired and unpaired T-Tests.

Also, See Student’s T-Test–for when samples are <30 in size.

When the sample groups are not independent, the appropriate method to test for differences between the groups is known as a paired comparison test (or paired T-test or paired sample test).

Furthermore, use the Paired T-Distribution and Paired T-Test to identify if a change has significantly impacted a process. The Paired T-test is similar to a 1-sample T-test. A 1-sample T-Test compares one sample mean. At the same time, a paired T-Test calculates the difference between paired values and then performs a 1-sample T-Test on the differences.

When to use Paired T-tests and Paired T-distributions

The Paired T-distributions, Paired T-tests, Paired Comparison Tests, and Paired Sample Tests are parametric procedures. Paired Samples T-tests are used when the same group is tested twice. It is often used in “before and after” designs where the same individuals are measured before and after a treatment or improvement to see if changes occurred over time.

Paired T-Distribution and Paired T-Test Assumptions

• Two repeated or matched samples, in other words, must have a before and after design or matched pairs.
• Paired samples T-tests can have only two groups. Use ANOVA for more than two measures.
• The non-negotiable assumption for the paired samples is dependent variable must be quantitative.
• The paired T-Test assumes no extreme outliers.
• The dependent variable’s sampling distribution should be normally distributed. If the data is non-normal, use the Wilcoxon test.
• A dependent variable is a continuous variable measured at an interval or ratio.
• Unlike the independent samples T-Test, there is no assumption for homogeneity of variance with a paired sample T-Test.

What is the Hypothesis of the Paired T-test?

• The Null Hypothesis for a paired T-test is that the average difference between the two population means is zero (0). In other words, there is no significant difference between the two population means.
• The Alternative Hypothesis for a paired T-Test–there is a significant difference between the two population means.

How to conduct a Paired T-test

• Establish the Null Hypothesis and Alternative Hypothesis
• Determine the significance level
• Calculate the difference between each observation in the two groups
• Then, compute the mean difference (x̅ – µ)
• Calculate the standard deviation of differences (s) and then calculate the standard error, i.e., s/√n (where n is the sample size)
• Compute the T-Statistic, t= (x̅ – µ)/ s/√n
• Determine t critical value with n-1 degrees of freedom
• Finally, interpret the result. If the test statistic falls in the critical region, reject the null hypothesis.

Hypothesis Testing

A Tailed Hypothesis is an assumption about a population parameter. The assumption may or may not be accurate. One-tailed Hypothesis is a test of a hypothesis where the area of rejection is only in one direction. Two-tailed tests measure against the alternative that there is a significant difference between the two population means. The selection of one or Two-tailed tests depends upon the problem.

Example of Two-tailed Paired T-test

Example: Two operators check the same dimension on the same sample of 10 parts. Below are the results. Is there a significant operator measurement error? Test at the 5% significance level.

Solution Details:
We need to calculate the T-Statistic value using t= (x̅ – µ)/ s/√n and then compare it to a table value t _critical.

• H₀ =There is no significant measurement error between the two operators.
• H₁ =There is a significant measurement error between the two operators.

• n=10
• DF (degrees of freedom) = n-1 ; 10-1 =9
• Significance level =5%

Calculate the difference between each observation in the two groups

Compute the mean difference (x̅ – µ)

• µ (mean of first test) = 59.5 ; take the average of the 10 data points in Op1
• x̅ (mean of new test) = 60.2 ; take the average of the 10 data points in Op2
• x̅-µ = 60.2 – 59.5 =0.7

Calculate the Standard Deviation of differences (s)

• D= -7 ; d²=73
• Use s_d = sqrt [ (Σ(d_i – d)² / (n – 1) ] where d_i is the difference for pair i, d is the sample mean of the differences, or s=( √ ((n*d²)-D²)/df) / √(n)
• s = (√ ((10*73) -(-7*-7)/9) / √(10) = 2.7508

Calculate the standard error

Standard error = s/√n = 2.7508 / √10 = 0.869

Compute the T-Statistic

t= (x̅ – µ)/ s/√n = (0.7) / 0.869 = 0.805

Determine t critical value with n-1 degrees of freedom

Since this is a Two-tailed test at an alpha of 5% t _critical = 2.262

Interpret the results

Compare t _statistic to t _critical 0.805 < 2.262.  In Hypothesis Testing, a critical value is a point on the test distribution compared to the test statistic to determine whether to reject the Null Hypothesis. The t calculated is not in the rejection region. Hence, we fail to reject the Null Hypothesis and say there is no difference between the two mean values.

Paired T 2 Tailed Template

Answer: Paired T-Test. It tests whether the average difference between the two measurements is statistically significant from zero.

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