Ed 602 - Lesson 11 - The independent t-test

Lesson 11 will consist of the following topics

• Text Assignment for Lesson 11

• Introduction to Two Sample Tests

• The independent t-test

• Example problem using the independent t-test

• Using the Excel Spreadsheet program to calculate the independent t-test

• Additional problem using the independent t-test

• Lesson 11 Assignment

• Lesson 11 Quiz

Text Assignment for Lesson 11

For lesson 11, read pages 123-129 in Practical Statistics for Educators, Third Edition by Ruth Ravid (2005, University Press of America)
or read pages 303-309 in Basic Statistics for Behavioral Science Research 2nd ed by Mary B. Harris (1998, Allyn and Bacon)
or read pages 191-195 in Practical Statistics for Educators, 2nd Edition by Ruth Ravid (2000, University Press of America)
or read pages 175-184 in Practical Statistics for Educators by Ruth Ravid (1994, University Press of America).

Introduction to Two Sample Tests

In our last lesson we looked at inferential statistics to test hypotheses concerned with comparing a single sample with some population parameter. We discussed two statistics to use with single samples.

1. We used the Z-test to compare a sample mean with the population mean when the population standard deviation was known.
2. We used the one-sample t-test to compare a sample mean with the population mean when the population standard deviation was not known.

In this lesson we are going to consider the first of two very important and widely used t-tests to compare the means of two groups or two samples. This is a very common problem to compare the means of, for example, the experimental group and the control group on some independent variable.

The two-sample t-tests we will consider are

1. The independent t-test which is used to compare two sample means when the two samples are independent of one another.
2. The non-independent or dependent t-test which is used for matched samples (where the two samples are not independent of one another as they are matched) and for pre-test/post-test comparisons where the pre-test and post-test are taken on the same group of subjects.

In this lesson we will consider the independent t-test, and in the next lesson we will consider the dependent t-test.

The independent t-test

The independent t-test, as we have already mentioned is used when we wish to compare the statistical significance of a possible difference between the means of two groups on some independent variable and the two groups are independent of one another.

The formula for the independent t-test is

where

is the mean for group 1,

is the mean for group 2,

is the sum of squares for group 1,

is the sum of squares for group 2,

n₁ is the number of subjects in group 1, and

n₂ is the number of subjects in group 2.

The sum of squares is a new way of looking at variance. It gives us an indication of how spread out the scores in a sample are. The t-value we are finding is the difference between the two means divided by their sum of squares and taking the degrees of freedom into consideration.

and

We can see that each sum of squares is the sum of the squared scores in the sample minus the sum of the scores quantity squared divided by the size of the sample (n).

So to calculate the independent-t value we need to know:

1. The mean for sample or group 1
2. The mean for sample or group 2
3. The summation X and summation X squared for group 1
4. The summation X and summation X squared for group 2
5. The sample size for group 1 (n₁)
6. The sample size for group 2 (n₂)

We also need to know the degrees of freedom for the independent t-test which is:

Let's do a sample problem using the independent t-test.

Example problem using the independent t-test

Research Problem: Job satisfaction as a function of work schedule was investigated in two different factories. In the first factory the employees are on a fixed shift system while in the second factory the workers have a rotating shift system. Under the fixed shift system, a worker always works the same shift, while under the rotating shift system, a worker rotates through the three shifts. Using the scores below determine if there is a significant difference in job satisfaction between the two groups of workers.

In this problem we see that we have two samples and the samples are independent of one another. We can see that the inferential statistic we need to use here is the independent t-test.

We can calculate the quantities we need to solve this problem as follows:

We can use the totals from this worksheet and the number of subjects in each group to calculate the sum of squares for group 1, the sum of squares for group 2, the mean for group 1, the mean for group 2, and the value for the independent t.

We now have the information we need to complete the six step statistical inference process for our research problem.

1. State the null hypothesis and the alternative hypothesis based on your research question.
   
   
   Note: Our problem did not state which direction of significance we will be looking for; therefore we will be looking for a significant difference between the two means in either direction.
2. Set the alpha level.
   
   Note: As usual we will set our alpha level at .05, we have 5 chances in 100 of making a type I error.
3. Calculate the value of the appropriate statistic. Also indicate the degrees of freedom for the statistical test if necessary.
   t = 2.209
   df = n₁ + n₂ - 2 = 11 + 11 - 2 = 20
   Note: We have calculated the t-value and will also need to know the degrees of freedom when we go to look up the critical values of t.
4. Write the decision rule for rejecting the null hypothesis.
   Reject H₀ if t is >= 2.086 or if t <= -2.086
   Note: To write the decision rule we need to know the critical value for t, with an alpha level of .05 and a two-tailed test. We can do this by looking at Appendix C (Distribution of t) on page 318 of the text book. Look for the column of the table under .05 for Level of significance for two-tailed tests, read down the column until you are level with 20 in the df column, and you will find the critical value of t which is 2.086. That means our result is significant if the calculated t value is less than or equal to -2.086 or is greater than or equal to 2.086.
5. Write a summary statement based on the decision.
   Reject H₀, p < .05, two-tailed
   Note: Since our calculated value of t (2.209) is greater than or equal to 2.086, we reject the null hypothesis and accept the alternative hypothesis.
6. Write a statement of results in standard English.
   There is a significant difference in job satisfaction betwen the two groups of workers.

Please send electronic mail to the course instructor if you have any questions about this lesson or other concerns.