1. © 2010 Pearson Prentice Hall. All rights reserved Chapter Hypothesis Tests Regarding a Parameter 10

2. © 2010 Pearson Prentice Hall. All rights reserved Section Hypothesis Tests for a Population Mean- Population Standard Deviation Unknown 10.3

3. © 2010 Pearson Prentice Hall. All rights reserved 10-104 Objectives 1.Test the hypotheses about a population mean with  unknown

4. © 2010 Pearson Prentice Hall. All rights reserved 10-105 To test hypotheses regarding the population mean assuming the population standard deviation is unknown, we use the t- distribution rather than the Z-distribution. When we replace  with s, follows Student’s t-distribution with n-1 degrees of freedom.

5. © 2010 Pearson Prentice Hall. All rights reserved 10-106 1.The t-distribution is different for different degrees of freedom. Properties of the t-Distribution

6. © 2010 Pearson Prentice Hall. All rights reserved 10-107 1.The t-distribution is different for different degrees of freedom. 2.The t-distribution is centered at 0 and is symmetric about 0. Properties of the t-Distribution

7. © 2010 Pearson Prentice Hall. All rights reserved 10-108 1.The t-distribution is different for different degrees of freedom. 2.The t-distribution is centered at 0 and is symmetric about 0. 3.The area under the curve is 1. Because of the symmetry, the area under the curve to the right of 0 equals the area under the curve to the left of 0 equals 1/2. Properties of the t-Distribution

8. © 2010 Pearson Prentice Hall. All rights reserved 10-109 4.As t increases (or decreases) without bound, the graph approaches, but never equals, 0. Properties of the t-Distribution

9. © 2010 Pearson Prentice Hall. All rights reserved 10-110 4.As t increases (or decreases) without bound, the graph approaches, but never equals, 0. 5.The area in the tails of the t-distribution is a little greater than the area in the tails of the standard normal distribution because using s as an estimate of  introduces more variability to the t-statistic. Properties of the t-Distribution

10. © 2010 Pearson Prentice Hall. All rights reserved 10-111 6.As the sample size n increases, the density curve of t gets closer to the standard normal density curve. This result occurs because as the sample size increases, the values of s get closer to the values of  by the Law of Large Numbers. Properties of the t-Distribution

11. © 2010 Pearson Prentice Hall. All rights reserved 10-112 Objective 1 Test hypotheses about a population mean with  unknown

12. © 2010 Pearson Prentice Hall. All rights reserved 10-113 Testing Hypotheses Regarding a Population Mean with σ Unknown To test hypotheses regarding the population mean with  unknown, we use the following steps, provided that: 1.The sample is obtained using simple random sampling. 2.The sample has no outliers, and the population from which the sample is drawn is normally distributed or the sample size is large (n≥30).

13. © 2010 Pearson Prentice Hall. All rights reserved 10-114 Step 1: Determine the null and alternative hypotheses. The hypotheses can be structured in one of three ways:

14. © 2010 Pearson Prentice Hall. All rights reserved 10-115 Step 2: Select a level of significance, , based on the seriousness of making a Type I error.

15. © 2010 Pearson Prentice Hall. All rights reserved 10-116 Step 3: Compute the test statistic which follows Student’s t-distribution with n-1 degrees of freedom.

16. © 2010 Pearson Prentice Hall. All rights reserved 10-117 Step 4: Use Table VI to determine the critical value using n-1 degrees of freedom. Classical Approach

17. © 2010 Pearson Prentice Hall. All rights reserved 10-118 Classical Approach Two-Tailed

18. © 2010 Pearson Prentice Hall. All rights reserved 10-119 Classical Approach Left-Tailed

19. © 2010 Pearson Prentice Hall. All rights reserved 10-120 Classical Approach Right-Tailed

20. © 2010 Pearson Prentice Hall. All rights reserved 10-121 Step 5: Compare the critical value with the test statistic: Classical Approach

21. © 2010 Pearson Prentice Hall. All rights reserved 10-122 Step 4: Use Table VI to estimate the P-value using n-1 degrees of freedom. P-Value Approach

22. © 2010 Pearson Prentice Hall. All rights reserved 10-123 P-Value Approach Two-Tailed

23. © 2010 Pearson Prentice Hall. All rights reserved 10-124 P-Value Approach Left-Tailed

24. © 2010 Pearson Prentice Hall. All rights reserved 10-125 P-Value Approach Right-Tailed

25. © 2010 Pearson Prentice Hall. All rights reserved 10-126 Step 5: If the P-value < , reject the null hypothesis. P-Value Approach

26. © 2010 Pearson Prentice Hall. All rights reserved 10-127 Step 6: State the conclusion.

27. © 2010 Pearson Prentice Hall. All rights reserved 10-128 Parallel Example 1: Testing a Hypothesis about a Population Mean, Large Sample Assume the resting metabolic rate (RMR) of healthy males in complete silence is 5710 kJ/day. Researchers measured the RMR of 45 healthy males who were listening to calm classical music and found their mean RMR to be 5708.07 with a standard deviation of 992.05. At the  =0.05 level of significance, is there evidence to conclude that the mean RMR of males listening to calm classical music is different than 5710 kJ/day?

28. © 2010 Pearson Prentice Hall. All rights reserved 10-129 Solution We assume that the RMR of healthy males is 5710 kJ/day. This is a two-tailed test since we are interested in determining whether the RMR differs from 5710 kJ/day. Since the sample size is large, we follow the steps for testing hypotheses about a population mean for large samples.

29. © 2010 Pearson Prentice Hall. All rights reserved 10-130 Solution Step 1: H 0 :  =5710 versus H 1 :  ≠5710 Step 2: The level of significance is  =0.05. Step 3: The sample mean is = 5708.07 and the sample standard deviation is s=992.05. The test statistic is

30. © 2010 Pearson Prentice Hall. All rights reserved 10-131 Solution: Classical Approach Step 4: Since this is a two-tailed test, we determine the critical values at the  =0.05 level of significance with n-1=45-1=44 degrees of freedom to be approximately -t 0.025 = -2.021 and t 0.025 =2.021. Step 5: Since the test statistic, t 0 =-0.013, is between the critical values, we fail to reject the null hypothesis.

31. © 2010 Pearson Prentice Hall. All rights reserved 10-132 Solution: P-Value Approach Step 4: Since this is a two-tailed test, the P-value is the area under the t-distribution with n-1=45-1=44 degrees of freedom to the left of -t 0.025 = -0.013 and to the right of t 0.025 =0.013. That is, P-value = P(t 0.013) = 2 P(t > 0.013). 0.50 < P-value. Step 5: Since the P-value is greater than the level of significance (0.05<0.5), we fail to reject the null hypothesis.

32. © 2010 Pearson Prentice Hall. All rights reserved 10-133 Solution Step 6: There is insufficient evidence at the  =0.05 level of significance to conclude that the mean RMR of males listening to calm classical music differs from 5710 kJ/day.

33. © 2010 Pearson Prentice Hall. All rights reserved 10-134 Parallel Example 3: Testing a Hypothesis about a Population Mean, Small Sample According to the United States Mint, quarters weigh 5.67 grams. A researcher is interested in determining whether the “state” quarters have a weight that is different from 5.67 grams. He randomly selects 18 “state” quarters, weighs them and obtains the following data. 5.70 5.67 5.73 5.61 5.70 5.67 5.65 5.62 5.73 5.65 5.79 5.73 5.77 5.71 5.70 5.76 5.73 5.72 At the  =0.05 level of significance, is there evidence to conclude that state quarters have a weight different than 5.67 grams?

34. © 2010 Pearson Prentice Hall. All rights reserved 10-135 Solution We assume that the weight of the state quarters is 5.67 grams. This is a two-tailed test since we are interested in determining whether the weight differs from 5.67 grams. Since the sample size is small, we must verify that the data come from a population that is normally distributed with no outliers before proceeding to Steps 1-6.

35. © 2010 Pearson Prentice Hall. All rights reserved 10-136 Assumption of normality appears reasonable.

36. © 2010 Pearson Prentice Hall. All rights reserved 10-137 No outliers.

37. © 2010 Pearson Prentice Hall. All rights reserved 10-138 Solution Step 1: H 0 :  =5.67 versus H 1 :  ≠5.67 Step 2: The level of significance is  =0.05. Step 3: From the data, the sample mean is calculated to be 5.7022 and the sample standard deviation is s=0.0497. The test statistic is

38. © 2010 Pearson Prentice Hall. All rights reserved 10-139 Solution: Classical Approach Step 4: Since this is a two-tailed test, we determine the critical values at the  =0.05 level of significance with n-1=18-1=17 degrees of freedom to be -t 0.025 = -2.11 and t 0.025 =2.11. Step 5: Since the test statistic, t 0 =2.75, is greater than the critical value 2.11, we reject the null hypothesis.

39. © 2010 Pearson Prentice Hall. All rights reserved 10-140 Solution: P-Value Approach Step 4: Since this is a two-tailed test, the P-value is the area under the t-distribution with n-1=18-1=17 degrees of freedom to the left of -t 0.025 = -2.75 and to the right of t 0.025 =2.75. That is, P-value = P(t 2.75) = 2 P(t > 2.75). 0.01 < P-value < 0.02. Step 5: Since the P-value is less than the level of significance (0.02<0.05), we reject the null hypothesis.

40. © 2010 Pearson Prentice Hall. All rights reserved 10-141 Solution Step 6: There is sufficient evidence at the  =0.05 level of significance to conclude that the mean weight of the state quarters differs from 5.67 grams.

41. © 2010 Pearson Prentice Hall. All rights reserved 10-142 Summary Which test to use? Provided that the population from which the sample is drawn is normal or that the sample size is large, if  is known, use the Z-test procedures from Section 10.2 if  is unknown, use the t-test procedures from this Section.