Question

To test $H_0: \mu = 0$ vs $H_1: \mu \neq 0$ when the population variance is known, which of the following test is used?

• A. $t$-test
• B. Chi-square test
• C. $Z$-test
• D. $F$-test

Hint:

Always check:
- Population variance known $\rightarrow$ Use $Z$-test.
- Population variance unknown $\rightarrow$ Use $t$-test (for small samples).

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
When testing hypotheses about a population mean, the choice of the test statistic depends primarily on whether the population variance $\sigma^2$ is known or unknown.

Step 2: Detailed Explanation:
Let us analyze the appropriate testing conditions:
- $Z$-test: Used to test hypotheses about a population mean when the population variance $\sigma^2$ is known (regardless of sample size), or when the sample size is large ($n \ge 30$) by using the sample variance as an approximation.
- $t$-test: Used to test hypotheses about a population mean when the population variance $\sigma^2$ is unknown and the sample size is small ($n < 30$).
- Chi-square ($\chi^2$) test: Used to test hypotheses about a single population variance or for testing independence/goodness-of-fit.
- $F$-test: Used to compare two population variances or within ANOVA to compare multiple means.
Since the population variance is explicitly specified as known, the $Z$-test is the correct statistical test to use.

Step 3: Final Answer
The correct option is (C).