Question

Which hypothesis test does not assume equal standard deviations/variance?

Answer 1

The hypothesis test that does not assume equal standard deviations/variance is the Welch's t-test.

Step-by-step explanation:

In statistical hypothesis testing, the standard t-test assumes equal variances between the two groups being compared. However, when variances are unequal, using the Welch's t-test is more appropriate. The Welch's t-test adjusts the degrees of freedom and provides a more accurate assessment of the significance of the differences between group means.

The formula for the Welch's t-test is given by:

`\[ t = \frac{(\bar{X}_1 - \bar{X}_2)}{\sqrt{(s_1^2)/(n_1) + (s_2^2)/(n_2)}} \]`

where
`\(\bar{X}_1\)` and
`\(\bar{X}_2\)` are the sample means,
`\(s_1^2\)` and
`\(s_2^2\)` are the sample variances, and
`\(n_1\)` and
`\(n_2\)` are the sample sizes. The Welch's t-test is particularly useful when dealing with unequal sample sizes and unequal variances, providing a more robust statistical analysis.

By using this test, researchers can make more accurate inferences about population parameters even when the assumption of equal variances is violated. This is crucial in scenarios where maintaining equal variances is not realistic or feasible, ensuring that the statistical analysis is both valid and reliable.