Question

Let $x_1, x_2, \ldots, x_{n_1}$ and $y_1, y_2, \ldots, y_{n_2}$ are two independent random samples from normal population with same variance. To test the hypothesis, the t-test is given by : [Here $\mu_1$ and $\mu_2$ are means of two samples].

• A. $\frac{(\bar{x} - \bar{y}) - (\mu_1 - \mu_2)}{\sqrt{\frac{1}{(n_1 + n_2 - 2)} \left[ \sum(x_i - \bar{x})^2 + \sum(y_i - \bar{y})^2 \right] \left( \frac{1}{n_1} + \frac{1}{n_2} \right)}}$
• B. $\frac{(\bar{x} - \bar{y}) - (\mu_1 + \mu_2)}{\sqrt{\frac{1}{(n_1 + n_2 - 2)} \left[ \sum(x_i - \bar{x})^2 + \sum(y_i - \bar{y})^2 \right] \left( \frac{1}{n_1} + \frac{1}{n_2} \right)}}$
• C. $\frac{(\bar{x} + \bar{y}) - (\mu_1 - \mu_2)}{\sqrt{\frac{1}{(n_1 + n_2 - 2)} \left[ \sum(x_i - \bar{x})^2 + \sum(y_i - \bar{y})^2 \right] \left( \frac{1}{n_1} + \frac{1}{n_2} \right)}}$
• D. $\frac{(\bar{x} + \bar{y}) + (\mu_1 - \mu_2)}{\sqrt{\frac{1}{(n_1 + n_2 - 2)} \left[ \sum(x_i - \bar{x})^2 + \sum(y_i - \bar{y})^2 \right] \left( \frac{1}{n_1} + \frac{1}{n_2} \right)}}$

Hint:

In hypothesis testing, the numerator is always (Observed Value - Expected Value). Since we are testing the difference between two groups, we look for the subtraction sign between the means.

Answer 1

The Correct Option is A

Concept: The independent two-sample t-test is used to determine if there is a significant difference between the means of two independent groups. When the populations are assumed to have the same variance, a "pooled" estimate of the variance is used.

Step 1: Determine the numerator structure.
The t-statistic measures the difference between the observed difference in sample means $(\bar{x} - \bar{y})$ and the hypothesized difference in population means $(\mu_1 - \mu_2)$. Therefore, the numerator must be: \[ (\bar{x} - \bar{y}) - (\mu_1 - \mu_2) \]

Step 2: Identify the Pooled Variance and Standard Error.
The denominator represents the estimated standard error of the difference between the means. For samples with equal variance, we use the pooled variance: \[ s_p^2 = \frac{\sum(x_i - \bar{x})^2 + \sum(y_j - \bar{y})^2}{n_1 + n_2 - 2} \] The standard error is then $s_p \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}$.

Step 3: Final Formula Construction.
Combining the steps, the full expression for the t-statistic is: \[ t = \frac{(\bar{x} - \bar{y}) - (\mu_1 - \mu_2)}{\sqrt{\frac{\sum(x_i - \bar{x})^2 + \sum(y_i - \bar{y})^2}{n_1 + n_2 - 2} \left( \frac{1}{n_1} + \frac{1}{n_2} \right)}} \] This matches the mathematical structure in Option (1).