Question

A two-independent sample t-test statistic based on sample sizes 15 and 18 with equal population variances follows

• A. $t$ distribution with 31 degrees of freedom
• B. $t$ distribution with 30 degrees of freedom
• C. $t$ distribution with 32 degrees of freedom
• D. $Z$ distribution with 33 degrees of freedom

Hint:

For a two-sample independent $t$-test, we lose exactly 2 degrees of freedom because we estimate two sample means ($\bar{X}_1$ and $\bar{X}_2$) to compute the pooled variance.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
When comparing the means of two independent samples with equal but unknown population variances, we use a two-sample student's $t$-test.
Key Formula or Approach:
The degrees of freedom ($df$) for a two-independent sample $t$-test with sample sizes $n_1$ and $n_2$ under the assumption of equal population variances is given by:
\[ df = n_1 + n_2 - 2 \]

Step 2: Detailed Explanation:
We are given:
Sample size of the first group, $n_1 = 15$
Sample size of the second group, $n_2 = 18$
Substitute these values into the degrees of freedom formula:
\[ df = 15 + 18 - 2 = 31 \]
Therefore, the test statistic follows a $t$ distribution with 31 degrees of freedom.

Step 3: Final Answer
The correct option is (A).