404 CHAPTER 8 Hypothesis Testing c. Use the sample data to construct a 95% confidence interval estimate of the proportion of zeros. What does the confidence interval suggest about the claim that the proportion of zeros equals 0.1? d. Compare the results from the critical value method, the P-value method, and the confidence interval method. Do they all lead to the same conclusion? 35. Power For a hypothesis test with a specified significance level a, the probability of a type I error is a, whereas the probability b of a type II error depends on the particular value of p that is used as an alternative to the null hypothesis. a. Using an alternative hypothesis of p 6 0.4, using a sample size of n = 50, and assuming that the true value of p is 0.25, find the power of the test. See Exercise 30 “Calculating Power” in Section 8-1. [Hint: Use the values p = 0.25 and pq>n = 10.25210.752>50.] b. Find the value of b, the probability of making a type II error. c. Given the conditions cited in part (a), find the power of the test. What does the power tell us about the effectiveness of the test? 36.Claim of “At Least” or “At Most” How do the following results change? a. Chapter Problem claim is changed to this: “At least 50% of Internet users utilize two-factor authentication to protect their online data.” b. Exercise 15 “News Media” claim is changed to this: “At most half of Americans prefer to watch the news rather than read or listen to it.” Key Concept Testing a claim about a population mean is one of the most important methods presented in this book. This section deals with the very realistic and commonly used case in which the population standard deviation s is not known. (There is a brief discussion of the procedure used when s is known, which is very rare.) Testing a Claim About M with S Not Known In reality, it is very rare that we test a claim about an unknown value of a population mean m but we somehow know the value of the population standard deviation s. The realistic situation is that we test a claim about a population mean and the value of the population standard deviation s is not known. When s is not known, we estimate it with the sample standard deviation s. From the central limit theorem (Section 6-4), we know that the distribution of sample means x is approximately a normal distribution with mean mx = m and standard deviation sx = s>1n, but if s is unknown, we estimate it with s>1n, which is used in the test statistic for a “t test.” This test statistic has a distribution called the Student t distribution. The requirements, test statistic, P-value, and critical values are summarized in the Key Elements box that follows. Equivalent Methods For the t test described in this section, the P-value method, the critical value method, and the confidence interval method are all equivalent in the sense that they all lead to the same conclusions. 8-3 Testing a Claim About a Mean
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