Question

What factors affect statistical power when employing a z-test for a known population mean standard deviation and one-tailed hypotheses?

A) Sample size
B) Significance level
C) Effect size
D) Variability in the data

Answer 1

Factors that affect the power of a z-test include sample size, significance level, effect size, and variability in the data. For a hypothesis test at a 5% significance level, if population standard deviations are known, a z-test is used, with the random variable being the difference between sample and population means.

Step-by-step explanation:

The factors that affect statistical power when employing a z-test for a known population mean standard deviation and one-tailed hypotheses include:

• Sample size (A): Larger sample sizes generally increase the power of a test, as they tend to produce more accurate estimates of the population mean and reduce sampling error.
• Significance level (B): A higher significance level (α), such as 0.05, means that you are more willing to reject the null hypothesis when it is actually true (Type I error), which increases power. However, this also increases the chance of obtaining a false positive result.
• Effect size (C): The larger the difference between the hypothesized population mean and the true population mean (effect size), the higher the power of the test. A small effect size may require a larger sample to detect with the same power.
• Variability in the data (D): Less variability (or standard deviation) in the population increases power because it means that any observed difference is more likely to stem from a true effect rather than from random fluctuation.

To conduct an appropriate hypothesis test at the 5 percent significance level (b), we must first establish if the population standard deviations are known or unknown. If they are known and other conditions (such as normality of the data and independent samples) are met, we use a z-test. In the scenario described, the random variable would be the difference between the sample mean and the known population mean, measured in units of the known standard deviation.

Note: When mentioned that a high power is favorable, it refers to the probability (1 - β) that the test will correctly reject a false null hypothesis, indicating a strong test.