Paired T test with solved example

1.             Introduction

        A paired t-test compares the means of two related groups to determine if there is a statistically significant difference between them. This test is used when the same subjects are measured twice, such as before and after a treatment. By analyzing the differences between paired observations, the paired t-test accounts for variability within subjects, making it a powerful tool for detecting changes or effects in experiments and repeated-measures studies.

2.             When to use paired t test

A paired t-test is used to compare the means of two related groups to determine if there is a statistically significant difference. This test is applicable when the same subjects are measured twice or when pairs of related subjects are matched. Use a paired t-test in the following scenarios:

1.        The same subjects are measured twice, such as before and after treatment.

2.        Pairs of related subjects are matched based on specific criteria.

3.    The objective is to detect significant changes within subjects by analyzing differences between paired observations.

3.     Assumptions of paired t-test

1.   Paired Observations: The data consists of pairs of related observations (e.g., measurements before and after treatment on the same subjects).

2.  Continuous Data: The differences between paired observations are continuous and measured on an interval or ratio scale.

3.     Normality: The differences between the paired observations are approximately normally distributed.

4.        Independence: The pairs are independent of each other.

5.      Scale of Measurement: The data should be measured at least at the interval level, allowing for the calculation of meaningful differences.

4.     How is paired t-test is different from two sample independent t-test

The paired t-test compares the means of two related groups, focusing on differences within the same subjects, assuming paired observations. In contrast, the two-sample independent t-test compares the means of two independent groups, assuming observations are independent and unrelated. The paired t-test controls for within-subject variability, while the independent t-test assesses differences between separate groups, each with its own distinct subjects.

 

6.     Solved example

The data of the following table indicates the milk production of 12 cows for evening and morning milking. Test that the milk production of both the time is same.

Milk yield in liter

Cows

Morning (Xi)

Evening (Yi)

Cows

Morning (Xi)

Evening (Yi)

1

4.5

4.0

7

5.0

4.5

2

5.6

4.5

8

7.5

7.5

3

7.5

7.5

9

10.5

10.0

4

8.0

7.6

10

7.0

7.0

5

8.0

5.5

11

10.0

9.5

6

8.5

8.5

12

8.5

8.5






Same problem analysis in Agri Analyze:

The best part of performing analysis with Agri Analyze is that auto interpretation along with assumption testing.

Step 1: Open link https://agrianalyze.com/OneSampleTTest.aspx (For first time users free registration is mandatory)

Step 2: Prepare data in csv file as shown below:

Link of the dataset

Step3: Upload data, add level of significance and average difference, add variable name and category type


Add level of Significance: we have kept 0.05 i.e. 5%; 

Add variable name : Milk production in Liter

Category Type: As we are comparing milk production in Morning and Evening we have wrote "Time of Milking"

Step4: Click on the submit and pay nominal fee and get the output along with interpretation in report.

Output:




Link of the output report
Reference:
Gupta, S. C., & Kapoor, V. K. (2020). Fundamentals of mathematical statistics. Sultan Chand & Sons.

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