366 CHAPTER 7 Hypothesis Testing with One Sample Hypothesis Testing Using a P-Value In auto racing, a pit stop is where a racing vehicle stops for new tires, fuel, repairs, and other mechanical adjustments. The efficiency of a pit crew that makes these adjustments can affect the outcome of a race. A pit crew claims that its mean pit stop time (for 4 new tires and fuel) is less than 13 seconds. A random sample of 32 pit stop times has a sample mean of 12.9 seconds. Assume the population standard deviation is 0.19 second. Is there enough evidence to support the claim at a = 0.01? Use a P@value. SOLUTION Because s is known 1s = 0.192, the sample is random, and n = 32 Ú 30, you can use the z@test. The claim is “the mean pit stop time is less than 13 seconds.” So, the null and alternative hypotheses are H0: m Ú 13 seconds and Ha: m 6 13 seconds. (Claim) The level of significance is a = 0.01. The standardized test statistic is z = x - m s 1n Because s is known and n Ú 30, use the z-test. = 12.9 - 13 0.19 232 Assume m = 13. ≈ -2.98. Round to two decimal places. Using Table 4 in Appendix B, the area corresponding to z = -2.98 is 0.0014. Because this test is a left-tailed test, the P@value is equal to the area to the left of z = -2.98, as shown in the figure at the left. So, P = 0.0014. Because the P@value is less than a = 0.01, you reject the null hypothesis. You can check your answer using technology, as shown below. Note that the P@value differs slightly from the one you found due to rounding. STATCRUNCH One sample Z summary hypothesis test: μ : Mean of population H0 : μ = 13 HA : μ 6 13 Standard deviation = 0.19 Hypothesis test results: Mean n Sample Mean Std. Err. Z-Stat P-value μ 32 12.9 0.033587572 -2.9772917 0.0015 Interpretation There is enough evidence at the 1% level of significance to support the claim that the mean pit stop time is less than 13 seconds. TRY IT YOURSELF 4 Homeowners claim that the mean speed of automobiles traveling on their street is greater than the speed limit of 35 miles per hour. A random sample of 100 automobiles has a mean speed of 36 miles per hour. Assume the population standard deviation is 4 miles per hour. Is there enough evidence to support the claim at a = 0.05? Use a P@value. Answer: Page A40 EXAMPLE 4 z 1 2 3 −1 −2 −3 0 z = −2.98 The area to the left of z = −2.98 is P = 0.0014. Left-Tailed Test
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