t-distribution Summary MBA 605, Business Analytics Donald D. Conant, Ph.D.
Types of t-tests There are several types of t-test. In this course we discuss three. The single-sample t-test The two-sample t-test assuming equal variances The two-sample t-test assuming unequal variances
The Single-Sample t-test A key characteristic of the single-sample t-test is the number of known variances. In a single-sample t-test only one variance is provided. There are generally two ways in which the problem is structured. In the first the hypothesized value (µ), the sample mean ( ), and the standard deviation of the sample (s) are provided. In the second the hypothesized value (µ), and the sample data (ie., 12, 14, 8, 11, 13) are provided. In this case the sample mean ( ), and the standard deviation of the sample (s) must be calculated. = 11.6, s = 2.3. In both cases the t-test method is the same. The second example merely adds an additional step.
The Single-Sample t-test To solve a problem the following information must be calculated using µ,, s, and the sample size (n) The critical value of t (t cv ) The formulas for and t are provided above. To solve for t cv additional information is needed.
The Single Sample t-test To solve for the critical value of t (t cv ) the following is needed. The Number of tails (an equals sign in H 0 indicates a 2-tailed test, less-than or greater-than sign indicates a 1-tailed test. The researcher must decide on the alpha (α) value. This determines the probability of committing a Type I error (rejecting H 0 when H 0 is true). Common α values in social science research are.05,.01, and.001. Excel s T.INV() function is used to find t cv.
The Single Sample t-test The T.INV() function requires the probability and the degrees of freedom. Probability = α/# of tails Degrees of freedom = the sample size (n) 1 For example, for a 2-tailed test with an α of.05 and a sample size of 5, t cv = T.INV(.05/2,5-1) = -2.776 A 2-tailed test requires an additional step. The calculation above determines the boundary of one tail. The following calculation is necessary to set the boundary of the other tail. t cv = T.INV(1-.05/2,5-1) = 2.776
The Single Sample t-test If the calculated t-value falls between -2.776 and +2.776 then fail to reject H 0. Otherwise reject H 0.
Confidence Interval (CI) When conducting single sample t-tests it is often useful to know the range of mean values for which the means would be considered equal. This is referred to as the Confidence Interval (CI). The CI consists of an Upper Bound and a Lower Bound. These are calculated using the following two formulas.
Confidence Interval (CI) In our example: = 11.6, tcv = 2.776, and = 1.02 Therefore, Upper Bound = 11.6 + (2.776)(1.02) = 11.6 + 2.83 = 14.43 Lower Bound = 11.6 (2.776)(1.02) = 11.6 2.83 = 8.77 This indicates that any hypothesized value () that is greater than or equal to 8.77 and less than or equal two 14.43 will satisfy the assumption that = µ, thus fail to reject H 0.
Two Sample t-tests A key characteristic of a two sample t-test is the number of known variances. In a two sample t-test has two variances, one for each sample. Generally two data samples are provided. Before proceeding with a two sample t-test an F-test must be conducted to determine if the variances of the two samples are equal or unequal. For the F-test, H 0 : The Excel F-Test Two-Sample for Variances is used to test the samples.
Two Sample t-test For example, if there were two groups of data Group 1: 16, 20, 10, 15, 8, 19, 14, 15 Group 2: 15, 18, 13, 10, 12, 16, 11, 12 From the Data tab in Excel select the Data Analysis add-in If it s not present select the green FILE tab, Options, find Manage: Excel Addins near the bottom and click on Go Place a check by Analysis Toolpack and Solver Add-in and click on OK. Select the F-Test Two-Sample for Variances Select the two groups, indicate the alpha level, indicate a destination for the output, and click on OK.
Two Sample t-test The results will look like this: Variable 1 Variable 2 Mean 14.625 13.375 Variance 16.554 7.411 Observations 8 8 df 7 7 F 2.234 P(F<=f) one tail 0.155 F Critical one tail 3.787 The only value of concern is P(F<=f) one-tail. If it is greater than or equal to the selected alpha value assume the variances are equal. Thus, fail to reject H 0.
Two Sample t-test Assuming the variances are equal, from the Data tab in Excel select Data Analysis, select the t-test: Two-Sample Assuming Equal Variances, and click on OK. Select the two groups, indicate the alpha level, indicate a destination for the output, and click on OK.
Two Sample t-test The results will look like this: For a two-tail test you are only concerned with P(T<=t) two-tail. For a one-tail test you are only concerned with P(T<=t) one-tail. In both instances if the P value is greater than the selected alpha value assume the means are equal. Thus, fail to reject H 0. Variable 1 Variable 2 Mean 14.625 13.375 Variance 16.554 7.411 Observations 8 8 Pooled Variance 11.982 Hypothesized Mean Diff. 0 df 14 t Stat 0.722 P(T<=t) one tail 0.241 t Critical one tail 1.761 P(T<=t) two tail 0.482 t Critical two tail 2.145