Which of the following is not true when conducting hypothesis testing with independent samples?

• A. The populations from which the samples are drawn should ideally be $normally$ distributed, especially for small sample sizes.
• B. The observations within each sample must be independent of each other.
• C. The two samples must have $equal$ sample sizes.
• D. The samples must be randomly selected from their respective populations.

Answer: (C)

Verified step by step guidance

1. Step 1: Understand the assumptions of hypothesis testing with independent samples. These typically include: the populations should be normally distributed (especially important for small samples), observations within each sample must be independent, and samples should be randomly selected from their populations.
2. Step 2: Recognize that equal sample sizes are not a strict requirement for conducting hypothesis tests with independent samples. While equal sizes can simplify calculations and improve power, many tests (like the two-sample t-test) can handle unequal sample sizes.
3. Step 3: Review each statement given in the problem and compare it against the standard assumptions: (a) normality of populations, (b) independence of observations within samples, (c) equal sample sizes, and (d) random sampling.
4. Step 4: Identify that the statement 'The two samples must have equal sample sizes' is not true as a general requirement for hypothesis testing with independent samples.
5. Step 5: Conclude that the other statements are true assumptions or conditions necessary for valid hypothesis testing, while equal sample sizes are not mandatory.