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Chapter 9: Problem 55

Define the power of a statistical test. As the alternative value of \(\mu\) gets farther from \(\mu_{0}\), how is the power affected?

Short Answer

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Answer: As the alternative value of 饾渿 gets farther from 饾渿0, the power of the statistical test increases. This is because the difference between the null hypothesis and the alternative hypothesis becomes more significant, making it easier to reject the null hypothesis when it should be rejected and reducing the likelihood of committing a Type II error.

Step by step solution

01

Definition of Power of a Statistical Test

The power of a statistical test is the probability of correctly rejecting the null hypothesis when it is false. In other words, it is the probability of not committing a Type II error. A Type II error occurs when we fail to reject the null hypothesis when it should have been rejected. Mathematically, the power of a test can be defined as: Power = 1 - \(\beta\) where \(\beta\) represents the probability of committing a Type II error.
02

Alternative Value of \(\mu\) Getting Farther from \(\mu_0\)

As the alternative value of \(\mu\) gets farther from \(\mu_0\), the difference between the null hypothesis and the alternative hypothesis becomes more significant. This means that the probability of rejecting the null hypothesis when it is false increases, as the evidence pointing towards an alternative result becomes stronger.
03

Effect on Power

The power of a statistical test is affected positively as the alternative value of \(\mu\) gets farther from \(\mu_0\). Since the power of a test is the probability of correctly rejecting the null hypothesis when it is false, a larger difference between the null hypothesis and the alternative hypothesis makes it easier to reject the null hypothesis when it should be rejected. Consequently, the probability of not committing a Type II error increases, and the power of the test also increases. In summary, as the alternative value of \(\mu\) gets farther from \(\mu_0\), the power of the statistical test increases, making it more likely to correctly reject the null hypothesis when it is false.

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Most popular questions from this chapter

Early Detection of Breast Cancer Of those women who are diagnosed to have early-stage breast cancer, one-third eventually die of the disease. Suppose a community public health department instituted a screening program to provide for the early detection of breast cancer and to increase the survival rate \(p\) of those diagnosed to have the disease. A random sample of 200 women was selected from among those who were periodically screened by the program and who were diagnosed to have the disease. Let \(x\) represent the number of those in the sample who survive the disease a. If you wish to detect whether the community screening program has been effective, state the null hypothesis that should be tested. b. State the alternative hypothesis. c. If 164 women in the sample of 200 survive the disease, can you conclude that the community screening program was effective? Test using \(\alpha=.05\) and explain the practical conclusions from your test. d. Find the \(p\) -value for the test and interpret it.

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