468 Chapter 9 Hypothesis Testing b. Find the power of the test against the alternative ��1 = 1.05. 0.9974 Tech: 0.9973 c. Find the power of the test against the alternative ��1 = 1.02 if the test is made at level �� = 0.01. 0.2912 Tech: 0.2897 d. Find the power of the test against the alternative ��1 = 1.05 if the test is made at level �� = 0.01. 0.9821 Tech: 0.9823 15. Tire lifetimes: Refer to Exercise 11. A test of the hypotheses H0 : �� = 50,000 versus H1: �� ≠ 50,000 will be made at the �� = 0.05 level of significance. Assume that the standard deviation of tire lifetimes is �� = 5000 and that the sample size is n = 100. Find the power of the test against the alternative ��1 = 49,000. 0.5160 16. Coffee beans: Refer to Exercise 12. A test of the hypotheses H0 : �� = 10 versus H1: �� ≠ 10 will be made at the �� = 0.05 level of significance. Assume that the standard deviation of moisture content is �� = 5.0 and that the sample size is n = 50. Find the power of the test against the alternative ��1 = 12. 0.8079 Tech: 0.8074 17. SAT scores: Refer to Exercise 13. The admissions officer will perform a test of the hypotheses H0 : �� = 500 versus H1: �� ≠ 500 at the �� = 0.01 level of significance. Assume that the population standard deviation is �� = 116 and that the sample size is n = 100. Find the power of the test against the alternative ��1 = 550. 0.9582 Tech: 0.9586 18. Watch your cholesterol: Refer to Exercise 14. A test of the hypotheses H0 : �� = 1 versus H1: �� ≠ 1 will be made at the �� = 0.01 level of significance. Assume that the population standard deviation is �� = 0.2 and that the sample size is n = 314. Find the power of the test against the alternative ��1 = 0.98. 0.2119 Tech: 0.2107 Extending the Concepts The power for a test of a population proportion can be found by a method similar to that for a population mean. 19. Power for test of a proportion: A marketing firm samples 150 residents of a certain town to determine the sample proportion ̂p that have seen a new advertisement. A test of the hypotheses H0 : p = 0.4 versus H1: p < 0.4 will be performed, using a significance level of �� = 0.05. a. Find the critical value z��. 1.645 b. Find the value of ̂p whose z-score is −z��. Call this value p∗. Note that the population standard deviation is �� = √ p0(1 − p0). 0.3342 c. We want to find the power against the alternative p1 = 0.3. The power is the area to the left of p∗ under the assumption that the true proportion is p1 = 0.3. Find the power. Note that the standard deviation is now �� = √ p1(1 − p1). 0.8186 Tech: 0.8197 Answers to Check Your Understanding Exercises for Section 9.7 1. 0.7611 Tech: 0.7618 2. 0.9962 Chapter 9 Summary Section 9.1: A hypothesis test involves a null hypothesis, H0, which makes a statement about one or more population parameters, and an alternate hypothesis, which contradicts H0. We begin by assuming that H0 is true. If the data provide strong evidence against H0, we then reject H0 and believe H1. A Type I error occurs when a true null hypothesis is rejected. A Type II error occurs when a false H0 is not rejected. Section 9.2: We follow one of two methods in performing a hypothesis test. In the critical value method, we choose a significance level ��, then find a critical region. We reject H0 if the test statistic falls inside the critical region. The probability of a Type I error is ��, the significance level of the test. In the P-value method, we compute a P-value, which is the probability of observing a value for the test statistic that is as extreme as or more extreme than the value actually observed, under the assumption that H0 is true. The smaller the P-value, the stronger the evidence against H0. If we want to make a firm decision about the truth of H0, we choose a significance level �� and reject H0 if P ≤ ��. When testing a hypothesis about a population mean with the population standard deviation �� known, the test statistic, z, has a standard normal distribution. If the sample size is not large, the population must be approximately normal. We can check normality with a boxplot or dotplot. Statistical significance is not the same as practical significance. When a result is statistically significant, we can conclude only that the true value of the parameter is different from the value specified by H0. We cannot conclude that the difference is large enough to be important. When presenting the results of a hypothesis test, it is important to state the P-value or the value of the test statistic, so that others can decide for themselves whether to reject H0. It isn’t enough simply to state whether or not H0 was rejected. Section 9.3: When testing a hypothesis about a population mean with the population standard deviation �� unknown, the test statistic has a Student’s t distribution. The number of degrees of freedom is 1 less than the sample size. The population must be approximately normal, or the sample size must be large (n > 30). We can check normality with a boxplot or dotplot. Section 9.4: When testing a hypothesis about a population proportion, the test statistic is z. The sample proportion must be approximately normal. We check this by requiring that both np0 and n(1 − p0) are at least 10.
navidi_monk_elementary_statistics_2e_ch7-9
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