Computing the t statistic for a 2-sample t test

Question

A teacher performs a hypothesis test to determine whether the mean test score for girls on an aptitude test differs from the mean test score for boys. She selects random samples of 50 girls and 60 boys. The sample standard deviations are 27.5 for the group of girls and 24.3 for the group of boys. The difference between the means is 8.1. What is the calculated t-value?

Answer 1

Comment continued: I'll show answers from software and let you use the appropriate formulas to see if you get the same answers. Neither version of the test shows a significant difference between scores of boys and girls. (Computer output is slightly edited for relevance.)

The Welch test is preferred unless there is strong prior evidence that population variances are equal, but in this case the difference between the two tests is slight. Because $n_1 \ne n_2,$ the absolute values of the t statistic are different between the pooled and Welch t tests. The degrees of freedom for the Welch test are often smaller (never larger).

From Minitab statistical software: Pooled t test; $\bar X_G > \bar X_B.$

Two-Sample T-Test

Sample   N   Mean  StDev  SE Mean
1       50  108.1   27.5      3.9
2       60  100.0   24.3      3.1


Difference = μ (1) - μ (2)
Estimate for difference:  8.10

T-Test of difference = 0 (vs ≠): T-Value = 1.64  P-Value = 0.104  DF = 108
Test uses Pooled StDev = 25.8011

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Welch test; $\bar X_G < \bar X_B.$

Two-Sample T-Test

Sample   N   Mean  StDev  SE Mean
1       50  100.0   27.5      3.9
2       60  108.1   24.3      3.1


Difference = μ (1) - μ (2)
Estimate for difference:  -8.10

T-Test of difference = 0 (vs ≠): T-Value = -1.62  P-Value = 0.108  DF = 98