Question

what conditions are necessary in order to use the t-test for testing the difference between two population means?

choose the correct answer below.

a. the population standard deviations are unknown. the populations are normally distributed or each sample size is at least 30.

b. the population standard deviations are unknown. the samples are randomly selected and independent.

c. the population standard deviations are unknown. the samples are randomly selected and independent. the populations are normally distributed or each sample size is at least 30.

d. the population standard deviations are known. the samples are randomly selected and independent. the populations are normally distributed or each sample size is at least 30.

Explanation:

To use the t - test for the difference between two population means, the following conditions must be met:

1. Population standard deviations ($\sigma_1$ and $\sigma_2$) are unknown (if they were known, we would use a z - test).
2. The samples must be randomly selected (to ensure representativeness of the populations) and independent (the selection of one sample does not affect the selection of the other sample).
3. The populations should be normally distributed or each sample size $n_1,n_2$ is at least 30 (by the Central Limit Theorem, for sample sizes $n\geq30$, the sampling distribution of the sample mean is approximately normal even if the population is not normally distributed).

• Option A: Missing the conditions that samples are randomly selected and independent.
• Option B: Missing the condition about the distribution of populations or sample size.
• Option D: Incorrectly states that population standard deviations are known (t - test is for unknown population standard deviations).

Answer:

C. The population standard deviations are unknown. The samples are randomly selected and independent. The populations are normally distributed or each sample size is at least 30.