Question

choose the correct answer below.
a. the population standard deviations are unknown. the populations are normally is at least 30.
b. the population standard deviations are unknown. the samples are randomly se
c. the population standard deviations are unknown. the samples are randomly se populations are normally distributed or each sample size is at least 30.
d. the population standard deviations are known. the samples are randomly sel populations are normally distributed or each sample size is at least 30.

Explanation:

To determine the correct option, we analyze the conditions for two - sample t - test (since population standard deviations are unknown, z - test is not used). The key conditions are:

1. Population standard deviations ($\sigma_1,\sigma_2$) are unknown.
2. Samples are randomly selected (independent samples, though not fully visible in options but a key condition).
3. Populations are normally distributed or sample sizes are at least 30 (Central Limit Theorem applies for large samples when population distribution is unknown).

• Option A: Mentions population standard deviations unknown and population normal or sample size at least 30? Wait, no, the description in A seems incomplete. But looking at option C: It states population standard deviations are unknown, samples are randomly selected, and populations are normally distributed or each sample size is at least 30. This matches the t - test conditions.
• Option B: Incomplete description, missing the normal population or large sample size condition.
• Option D: Says population standard deviations are known, which is wrong as we are in a situation where they are unknown (since we are considering t - test - like conditions here).

Answer:

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