372 CHAPTER 7 Hypothesis Testing with One Sample Hypothesis Testing Using Rejection Regions A researcher claims that the mean annual cost of raising a child between the ages of 1 and 3 in the U.S. is $13,350. In a random sample of families in the U.S., the mean annual cost of raising a child between the ages of 1 and 3 is $13,186. The sample consists of 1000 parents who had a child between the ages of 1 and 3. Assume the population standard deviation is $2750. At a = 0.10, is there enough evidence to reject the claim? (Adapted from U.S. News & World Report) SOLUTION Because s is known 1s = $27502, the sample is random, and n = 1000 Ú 30, you can use the z@test. The claim is “the mean annual cost is $13,350.” So, the null and alternative hypotheses are H0: m = $13,350 (Claim) and Ha: m ≠$13,350. Because the test is a two-tailed test and the level of significance is a = 0.10, the critical values are -z0 = -1.645 and z0 = 1.645. The rejection regions are z 6 -1.645 and z 7 1.645. The standardized test statistic is z = x - m s 1n Because s is known and n Ú 30, use the z-test. = 13,186 - 13,350 2750 21000 Assume m = $13,350. ≈ -1.89. Round to two decimal places. The figure shows the location of the rejection regions and the standardized test statistic z. Because z is in the rejection region, you reject the null hypothesis. z 1 2 3 −1 −2 −3 0 α= 0.05 z0 = 1.645 z ≈ −1.89 −z0 = −1.645 1 2 α= 0.05 1 2 α 1 − = 0.90 10% Level of Significance You can check your answer using technology, as shown below. MINITAB One-Sample Z N Mean SE Mean 90% CI for μ Z-Value P-Value 1000 13186.0 87.0 (13043.0, 13329.0) -1.89 0.059 μ: population mean of Sample Null hypothesis H0: μ = 13350 Known standard deviation = 2750 Alternative hypothesis H1: μ ≠ 13350 Interpretation There is enough evidence at the 10% level of significance to reject the claim that the mean annual cost of raising a child between the ages of 1 and 3 in the U.S. is $13,350. TRY IT YOURSELF 10 In Example 10, at a = 0.01, is there enough evidence to reject the claim? Answer: Page A41 EXAMPLE 10
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