In: Statistics and Probability
Explain the influence a level of significance and sample size has on hypothesis testing. Provide an example of the influence and explain how it impacts business decisions. In replies to peers, discuss whether you agree or disagree with the example provided and justify your response.
Power of a Hypothesis Test
The probability of not committing a Type II error is called the power of a hypothesis test.
Effect Size
To compute the power of the test, one offers an alternative view about the "true" value of the population parameter, assuming that the null hypothesis is false. The effect size is the difference between the true value and the value specified in the null hypothesis.
Effect size = True value - Hypothesized value
For example, suppose the null hypothesis states that a population mean is equal to 100. A researcher might ask: What is the probability of rejecting the null hypothesis if the true population mean is equal to 90? In this example, the effect size would be 90 - 100, which equals -10
Factors That Affect Power
The power of a hypothesis test is affected by three factors.
Test Your Understanding
Problem 1
Other things being equal, which of the following actions will reduce the power of a hypothesis test?
I. Increasing sample size.
II. Changing the significance level from 0.01 to 0.05.
III. Increasing beta, the probability of a Type II error.
(A) I only
(B) II only
(C) III only
(D) All of the above
(E) None of the above
Solution
The correct answer is (C). Increasing sample size makes the hypothesis test more sensitive - more likely to reject the null hypothesis when it is, in fact, false. Changing the significance level from 0.01 to 0.05 makes the region of acceptance smaller, which makes the hypothesis test more likely to reject the null hypothesis, thus increasing the power of the test. Since, by definition, power is equal to one minus beta, the power of a test will get smaller as beta gets bigger
Problem 2
Suppose a researcher conducts an experiment to test a hypothesis. If she doubles her sample size, which of the following will increase?
I. The power of the hypothesis test.
II. The effect size of the hypothesis test.
III. The probability of making a Type II error.
(A) I only
(B) II only
(C) III only
(D) All of the above
(E) None of the above
Solution
The correct answer is (A). Increasing sample size makes the hypothesis test more sensitive - more likely to reject the null hypothesis when it is, in fact, false. Thus, it increases the power of the test. The effect size is not affected by sample size. And the probability of making a Type II error gets smaller, not bigger, as sample size increases.
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