462 Chapter 9 Hypothesis Testing Is σ known? Yes No Yes Is the population approximately normal? Consult a statistician Is the population approximately normal? Consult a statistician Yes Yes No Yes Is n > 30? Is n > 30? No No No Use z-test (Section 9.2) Use t-test (Section 9.3) Use z-test (Section 9.2) Use t-test (Section 9.3) ∙ Population proportion: To perform a hypothesis test for a population proportion, use the method described in Section 9.4. ∙ Population standard deviation or variance: To perform a hypothesis test for a population standard deviation or variance, use the method described in Section 9.5. EXAMPLE 9.23 Determining which method to use Starting salaries for a random sample of 51 physicians had a mean of $103,000. Assume that the population standard deviation is $10,500. Can you conclude that the mean starting salary for physicians is greater than $100,000? Determine the type of parameter that is to be tested and perform the hypothesis test. Use the �� = 0.05 level of significance. Solution We are asked to perform a hypothesis test for the mean salary; this is a population mean. We consult the diagram to determine the correct method. We must first determine whether �� is known. We are told that the population standard deviation is $10,500. Therefore, �� = 10,500. We follow the ‘‘Yes’’ path. Next we must determine whether n > 30. The sample size is 51, so n > 30. We follow the ‘‘Yes’’ path, and find that we should use the z-test described in Section 9.2. To perform the test, we compute the value of the test statistic: z = 103,000 − 100,000 10,500∕ √ 51 = 2.04 Because the alternate hypothesis is H1: �� > 100,000, this is a right-tailed test. The P-value is the area under the normal curve to the right of z = 2.04. Using Table A.2, we find that the area to the left of z = 2.04 is 0.9793. The area to the right is therefore 1−0.9793 = 0.0207. The P-value is 0.0207. Because P < 0.05, we conclude that the mean starting salary is greater than $100,000. Check Your Understanding In Exercises 1–4, state which type of parameter is to be tested; then perform the hypothesis test. 1. In a simple random sample of 150 cars undergoing emissions testing, 23 failed the test. Can you conclude that the proportion of cars that fail the test is less than 20%? Use the �� = 0.05 level of significance. Proportion; do not reject H0. 2. A simple random sample of size 15 has mean ̄x = 27.72 and standard deviation s = 8.21. The population is approximately normally distributed. Can you conclude that the population mean differs from 35? Use the �� = 0.01 level of significance. Mean; reject H0.
navidi_monk_elementary_statistics_2e_ch7-9
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