Section 9.6 Determining Which Method to Use 461 0.1 ounce. The machine is moved to a new location. To determine whether the standard deviation has changed, ten cans are filled. Following are the amounts in the ten cans. Assume them to be a random sample from a normal population. 12.18 11.77 12.09 12.03 11.87 11.96 12.03 12.36 12.28 11.85 Perform a hypothesis test to determine whether the standard deviation differs from 0.1 ounce. Use the �� = 0.05 level of significance. What do you conclude? Reject H0. 20. Long-lasting drugs: One of the ways in which doctors try to determine how long a single dose of pain reliever will provide relief is to measure the drug’s half-life, which is the length of time it takes for one-half of the dose to be eliminated from the body. A report of the National Institutes of Health states that the standard deviation of the half-life of the pain reliever oxycodone is �� = 1.43 hours. Assume that a sample of 25 patients is given the drug, and the sample standard deviation of the half-lives was s = 1.5 hours. Assume the population is normally distributed. Can you conclude that the true standard deviation is greater than the value reported by the National Institutes of Health? Use the �� = 0.01 level of significance. Do not reject H0. Extending the Concepts The chi-square distribution is skewed, but as the number of degrees of freedom becomes large, the skewness diminishes. If the number of degrees of freedom, k, is large enough, the chi-square distribution is reasonably well approximated by a normal distribution with mean k and variance 2k. 21. Exact test: A sample of size 101 from a normal population has sample standard deviation s = 40. Use Table A.4 to find the exact critical values ��2 0.025 and ��2 0.975 to test H0 : �� = 30 versus H1: �� ≠ 30. Can you reject H0 at the �� = 0.05 level? Yes 22. Using the normal approximation: Refer to Exercise 21. Use the normal approximation to estimate the critical values ��2 0.025 and ��2 0.975. Using these critical values, can you reject H0 at the �� = 0.05 level? Yes A more accurate normal approximation to ��2 �� is given by ��2 �� ≈ 0.5 ( z�� + √ 2k − 1 )2 , where z�� is the z-score that has area �� to its right. 23. More accuracy: Refer to Exercise 21. Use the more accurate normal approximation to estimate the critical values ��2 0.025 and ��2 0.975. Using these critical values, can you reject H0 at the �� = 0.05 level? Yes Answers to Check Your Understanding Exercises for Section 9.5 1. ��2 = 23.457, critical value is 30.144. Do not reject H0. 2. ��2 = 2.396, critical value is 3.053. Reject H0. 3. ��2 = 75.000, critical values are 11.808 and 49.645. Reject H0. 4. ��2 = 1.960, critical values are 0.484 and 11.143. Do not reject H0. SECTION 9.6 Determining Which Method to Use Objectives 1. Determine which method to use when performing a hypothesis test Objective 1 Determine which method to use when performing a hypothesis test One of the challenges in performing a hypothesis test is to determine which method to use. The first step is to determine which type of parameter we are testing. There are three types of parameters about which we have learned to perform hypothesis tests: ∙ Population mean �� ∙ Population proportion p ∙ Population standard deviation �� or variance ��2 Once you have determined which type of parameter you are testing, proceed as follows: ∙ Population mean: There are two methods for performing a hypothesis test for a population mean, the z-test (Section 9.2) and the t-test (Section 9.3). To determine which method to use, we must determine whether the population is approximately normal, and whether the sample size is large (n > 30). The following diagram can help you make the correct choice.
navidi_monk_elementary_statistics_2e_ch7-9
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