# Paired Sample T-Test

Quick Links

- Definition of Paired Sample T-Test
- Hypotheses in Paired Sampled T-Test
- Assumptions in the Paired Sample T-Test
- Steps involved in Paired Sample T-Test
- Paired Sample T-Test symbols and Meaning
- Interpretation of Paired Sample T-Test
- Statistical Significance of the Paired Sample T-Test Results
- Paired Sample T-Test in SPSS
- The Chi-Square Statistic

**Definition of Paired Sample T-Test**

The paired sample t-test is also known as the dependent sample t-test. **It refers to a statistical model, step or procedure used to determine whether the mean difference between two sets of data observations is zero**.

When one is using a paired sample t-test, it is important to consider that **each data is evaluated and measured more than once**. You should use the paired sample t-test to measure the data test twice in order to get the desired pairs of observations.

Some of the common examples where a paired sample t-test has been used include in the application of case-control trials or studies as well as in the use of repeated-measures models.

**Practical application of paired sample t-test** also include a case where you want to measure, monitor and evaluate the levels of performance of employees in a company after their training and you use the paired sample t-test model or program and measure the performance of a sample of workers or the employees before and after completing the training program. You then use the paired sample t-test to evaluate and analyze the differences levels of performance among the workers after the training has been done.

**Hypotheses in Paired Sample T-Test**

Just like any other statistical programs, one should note that the paired sample t-test include two competing hypotheses. This is mainly the **null hypothesis** as well as the **alternative hypothesis**.

When using the **null hypothesis** you should assume and consider that the true mean difference between the paired samples or variables is zero. You should also note that the observable differences in the samples are caused by random variations.

However, **while using the alternative hypothesis**, you should understand that the true mean difference especially between the paired variables or samples is not equal to zero and may give a different value apart from zero. You should also understand that the alternative hypothesis can take various models and forms.

However, the correct form depends on the expected outcome to be achieved and the direction of the difference does to matter in the alternative hypothesis. This is why it is advisable that one uses a two-tailed hypothesis to avoid errors. *It is also worth noting that using a two-tailed hypothesis can help to increase the power of the test. *

One should also understand that the hypotheses in paired sample t-test are not about data. The hypotheses in paired sample t-test only point on the procedures that produce and include the correct data. In that sense, you should understand that t**he main goal of the process of hypothesis testing is to evaluate as well as to determine the hypothesis (null or alternative) with which the data are more consistent in the long run especially during data analysis**. Some of the common hypotheses in paired sample t-test are indicated below

- The
**null hypothesis (H0)**deduces and concludes that the actual mean difference (μd) is equal to zero. - The
**upper-tailed alternative hypothesis****(H1)**deduces and concludes that actual mean difference is greater than zero. - However, the
**lower-tailed alternative hypothesis (H1)**deduces and concludes that actual mean difference is less than zero - The
**two-tailed alternative hypothesis (H1)**deduces and concludes that actual difference is not equal to zero.

**Assumptions of the Paired Sample T-Test**

As a method used to determine the value of unknown parameters and variables or samples, paired sample t-test includes and covers a number of assumptions. **It considers deviations that are evident during testing** and these assumptions must be considered in the process of determining the paired sample t-test results. When you consider the assumptions it is easier to get valid and reliable data and results using the paired sample t-test method. The quality of the differences between two sets of values, or samples must also be considered in paired sample t-test.

The following are some of the common assumptions of the paired sample t-test:

- It should have a normal distribution curve
- It covers large data sets and this is important for easier analysis.
- It has dependent and independent variables or data sets
- The dependent sample should be continuous and occur at an interval ratio
- The observations witnessed do not depend on each other and can be analyzed differently
- The dependent model or sample should depict a normal distribution curve
- It lacks outliers and covers a variety of data which includes control trials
- The dependent variable does not include outliers in all data sets.
- It requires larger correlations and differences to provide valid and high-quality results
- The subjects number vary in different hypothesis

**Steps involved in Paired Sample T-Test**

There are various procedures involved in the paired sample t-test.

The **first** procedure or step involved in the paired sample t-test is the **calculation of the sample mean**. The sample mean must be calculated for you to get the undefined controlled value.

The **second step** involves the **calculation and analysis of the sample standard deviation**. The main aim of calculating the sample standard deviation in paired sample t-test is to get the undefined control value in a standardized form.

The **third step** involves the **calculation of the test statistic to get the control value of the t-test**. There is also the calculation of the probability of observing the test statistic under the null hypothesis. This value is obtained by comparing t to a t-distribution with (n − 1) degrees of freedom. This can be done by looking up the value in a table, such as those found in many statistical textbooks, or with statistical software for more accurate results.

All these steps require close analysis of the data and they can allow you to decide whether to accept or reject the hypothesis given. However, it should be noted that the steps are based on different symbols as shown below:

## Paired Sample T-Test symbols and meaning

D-refers to the differences between two paired samples

Di-refers to the ith observation in D

n-this is the sample size

d¯¯¯ -This is the sample mean of the differences

σ^-this refers to the sample standard deviation of the differences

T-This is the critical value of a t-distribution with (n − 1) degrees of freedom

t-this represents the t-statistic (t-test statistic) for a paired sample t-test

p-this is the p-value (probability value) for the t-statistic

**Interpretation of Paired Sample T-Test**

The interpretation of paired sample t-test requires the consideration of various factors. For example, one has to consider that the sample variables given can either be numeric or continuous. This is mainly because the sample t-test is based on the normal distribution.

To get the continuous data, you should provide different values, especially within a range of data such as in the analysis of the income level of a group of people or in determining the height or weight of products.** **You should also consider that the sample t-test gives both discrete and abstract values. This may include the data with low as well as medium and high ranges. This data can be well explained in a Likert-scale.

In interpreting the data, one should also consider their independence and normality. A good example of a normal data set in paired sample t-test is shown below

**Statistical Significance of the Paired Sample T-Test Results**

In order to determine the statistical significance of the paired sample t-test, one should consider the **practical** as well as **statistical significance**. The paired sample t-test includes a statistical significance and this is mainly determined by using the p-value.

In most cases, it is the **p-value gives the probability of observing the test results and findings**. This is common under the null hypothesis model. To get the correct data, one should know that **the lower the p-value, the lower the probability of obtaining a result, especially if the null hypothesis is true.**

When you see a low p-value you should understand that there is a low significance in the null hypothesis. However, the possibility that the null hypothesis is true and that we simply obtained on a very rare result can never be ruled out completely. This is why you have a role to provide the correct statistical significance.

In most cases, many scholars prefer adopting a p-value of about 0.05 or less to get the best statistical significance in the paired sample t-test. This means that a value of about 5% and below is good for getting the correct results in paired sample t-test, especially if the null hypothesis is true, relevant and accurate. One should also consider the practical significance which depends on the number and the size of the data.

**Paired Sample T-Test in SPSS**

The paired sample t-test can be done in SPSS. This is achieved by entering the data in SPSS and conducting the analysis. SPSS refers to a Statistical Package for the Social Sciences and it is mainly used for complex statistical data analysis by scholars. In paired sample t-test, the SPSS will use the data sets provided to compare means and provide the results. It is also true that the SPSS will highlight the variables provided. This is clearly depicted in the figure below:

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