A researcher wants to carry out a hypothesis test involving the mean for a sample
of n = 20. While the true value of the population standard deviation is unknown, the
researcher is reasonably sure that the population is normally distributed. Given this
information, which of the following statements would be correct?
A. The researcher should use the z-test because the population is assumed to be
normally distributed.
B. The t-test should be used because á and ì are unknown.
C. The t-test should be used because ó is unknown and the sample size is small.
D. The researcher should use the z-test because the sample size is less than 30.
How about C?
C. The t-test should be used because σ is unknown and the sample size is small.
The correct answer would be option C: The t-test should be used because σ (population standard deviation) is unknown and the sample size is small.
To understand why this is the correct answer, let's break down the options and reasoning:
A. The researcher should use the z-test because the population is assumed to be normally distributed.
This statement is not entirely correct. While it is assumed that the population is normally distributed, the z-test requires knowledge of the population standard deviation (σ). Since the standard deviation is unknown, a z-test cannot be utilized.
B. The t-test should be used because á and ì are unknown.
This statement is partly correct. The t-test is indeed used when the population standard deviation is unknown. However, the symbols "á" and "ì" are not typically used in the context of hypothesis testing. The correct symbols for the parameters in question are "α" (alpha) for the level of significance and "μ" (mu) for the population mean.
C. The t-test should be used because ó is unknown and the sample size is small.
This statement is correct. When the population standard deviation (σ) is unknown, we typically use the t-test. Furthermore, the t-test is particularly suitable when the sample size is small. Conventionally, a sample size less than 30 is considered small, which makes the t-test appropriate in this case.
D. The researcher should use the z-test because the sample size is less than 30.
This statement is incorrect. While it is true that a sample size less than 30 is often used as a rule of thumb for determining when to use the t-test, it is not the sole criterion. The primary deciding factor is whether the population standard deviation is known. Since the standard deviation is unknown in this scenario, the z-test cannot be used.
To sum up, when the researcher is reasonably sure that the population is normally distributed but the population standard deviation is unknown, the appropriate choice for hypothesis testing with a small sample size (n < 30) is the t-test.