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Chapter 10: Problem 65

The power of a test is influenced by the sample size and the choice of significance level. a. Explain how increasing the sample size affects the power (when significance level is held fixed). b. Explain how increasing the significance level affects the power (when sample size is held fixed).

Short Answer

Expert verified
Increasing the sample size increases the power of a test because a larger sample size provides a wider base to detect a false null hypothesis. Increasing the significance level likewise increases the power of a test since it lowers the threshold for rejecting the null hypothesis. However, a larger significance level also increases the risk of making a Type I error.

Step by step solution

01

Understanding Power in Hypothesis Testing

The power of a statistical test is the probability that the test will reject the null hypothesis when the alternative hypothesis is true. It's a way of avoiding Type II error (the failure to reject a false null hypothesis). Power is important in hypothesis testing and has a direct relation with sample size and significance level.
02

Effect of Increasing Sample Size

Part a is about discussing the effect of increasing sample size on the power of the test, while holding the significance level constant. Increasing the sample size, while holding the significance level constant, increases the power of the test. This is because a larger sample size allows for a better realm for the detection of a false null hypothesis. If the null hypothesis is false, an increased sample size improves the chances of picking sample values that are farther from the hypothesized value, making the difference more obvious and therefore leading to rejection of the null hypothesis.
03

Effect of Increasing Significance Level

Part b is about the effect of increasing the significance level on the power of the test, while keeping the sample size constant. A larger significance level means that we are more willing to accept the risk of a Type I error (rejecting a true null hypothesis). This makes it easier to reject the null hypothesis and thus increases the power of the test. However, increasing the significance level also increases the probability of a Type I error, which illustrates that we always have to balance power and reliability when deciding on the significance level.

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Most popular questions from this chapter

Duck hunting in populated areas faces opposition on the basis of safety and environmental issues. The San Luis Obispo Telegram-Tribune (June 18,1991 ) reported the results of a survey to assess public opinion regarding duck hunting on Morro Bay (located along the central coast of California). A random sample of 750 local residents included 560 who strongly opposed hunting on the bay. Does this sample provide sufficient evidence to conclude that the majority of local residents oppose hunting on Morro Bay? Test the relevant hypotheses using \(\alpha=.01\).

Much concern has been expressed in recent years regarding the practice of using nitrates as meat preservatives. In one study involving possible effects of these chemicals, bacteria cultures were grown in a medium containing nitrates. The rate of uptake of radio-labeled amino acid was then determined for each culture, yielding the following observations: \(\begin{array}{llllllll}7251 & 6871 & 9632 & 6866 & 9094 & 5849 & 8957 & 7978\end{array}\) \(\begin{array}{lllllll}7064 & 7494 & 7883 & 8178 & 7523 & 8724 & 7468\end{array}\) Suppose that it is known that the true average uptake for cultures without nitrates is 8000 . Do the data suggest that the addition of nitrates results in a decrease in the true average uptake? Test the appropriate hypotheses using a significance level of \(.10 .\)

Are young women delaying marriage and marrying at a later age? This question was addressed in a report issued by the Census Bureau (Associated Press, June 8 , 1991). The report stated that in 1970 (based on census results) the mean age of brides marrying for the first time was \(20.8\) years. In 1990 (based on a sample, because census results were not yet available), the mean was \(23.9\). Suppose that the 1990 sample mean had been based on a random sample of size 100 and that the sample standard deviation was \(6.4\). Is there sufficient evidence to support the claim that in 1990 women were marrying later in life than in 1970 ? Test the relevant hypotheses using \(\alpha=.01\). (Note: It is probably not reasonable to think that the distribution of age at first marriage is normal in shape.)

Let \(\pi\) denote the proportion of grocery store customers that use the store's club card. For a large sample \(z\) test of \(H_{0}: \pi=.5\) versus \(H_{a}: \pi>.5\), find the \(P\) -value associated with each of the given values of the test statistic: a. \(1.40\) d. \(2.45\) b. \(0.93\) e. \(-0.17\) c. \(1.96\)

The article referenced in Exercise \(10.34\) also reported that 470 of 1000 randomly selected adult Americans thought that the quality of movies being produced was getting worse. a. Is there convincing evidence that fewer than half of adult Americans believe that movie quality is getting worse? Use a significance level of \(.05\). b. Suppose that the sample size had been 100 instead of 1000 , and that 47 thought that the movie quality was getting worse (so that the sample proportion is still . 47 ). Based on this sample of 100 , is there convincing evidence that fewer than half of adult Americans believe that movie quality is getting worse? Use a significance level of \(.05\). c. Write a few sentences explaining why different conclusions were reached in the hypothesis tests of Parts (a) and (b).
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