1. 1 1 Slide © 2008 Thomson South-Western. All Rights Reserved Slides by JOHN LOUCKS St. Edward’s University

2. 2 2 Slide © 2008 Thomson South-Western. All Rights Reserved Chapter 9, Part A Hypothesis Testing Developing Null and Alternative Hypotheses Developing Null and Alternative Hypotheses Type I and Type II Errors Type I and Type II Errors Population Mean:  Known Population Mean:  Known Population Mean:  Unknown Population Mean:  Unknown The hypothesis tests in this chapter involve two population parameters: the population mean, μ 0, and the population proportion, p 0.

3. 3 3 Slide © 2008 Thomson South-Western. All Rights Reserved Developing Null and Alternative Hypotheses Hypothesis testing can be used to determine whether Hypothesis testing can be used to determine whether a statement about the value of a population parameter a statement about the value of a population parameter should or should not be rejected. The hypothesis tests should or should not be rejected. The hypothesis tests in this chapter involve two population parameters: the in this chapter involve two population parameters: the population mean, μ 0, and the population proportion, population mean, μ 0, and the population proportion, p 0. p 0. The null hypothesis, denoted by H 0, is a tentative The null hypothesis, denoted by H 0, is a tentative assumption about a population parameter. assumption about a population parameter. The alternative hypothesis, denoted by H a, is the The alternative hypothesis, denoted by H a, is the opposite of what is stated in the null hypothesis. opposite of what is stated in the null hypothesis. The alternative hypothesis is what the test is The alternative hypothesis is what the test is attempting to establish. attempting to establish.

4. 4 4 Slide © 2008 Thomson South-Western. All Rights Reserved n Testing Research Hypotheses Developing Null and Alternative Hypotheses The research hypothesis should be expressed as The research hypothesis should be expressed as the alternative hypothesis. the alternative hypothesis. The conclusion that the research hypothesis is true The conclusion that the research hypothesis is true comes from sample data that contradict the null comes from sample data that contradict the null hypothesis. hypothesis.

5. 5 5 Slide © 2008 Thomson South-Western. All Rights Reserved Developing Null and Alternative Hypotheses n Testing the Validity of a Claim Manufacturers’ claims are usually given the benefit Manufacturers’ claims are usually given the benefit of the doubt and stated as the null hypothesis. of the doubt and stated as the null hypothesis. The conclusion that the claim is false comes from The conclusion that the claim is false comes from sample data that contradict the null hypothesis. sample data that contradict the null hypothesis.

6. 6 6 Slide © 2008 Thomson South-Western. All Rights Reserved n Testing in Decision-Making Situations Developing Null and Alternative Hypotheses A decision maker might have to choose between A decision maker might have to choose between two courses of action, one associated with the null two courses of action, one associated with the null hypothesis and another associated with the hypothesis and another associated with the alternative hypothesis. Action may be taken if the alternative hypothesis. Action may be taken if the null hypothesis is rejected and action may be taken null hypothesis is rejected and action may be taken if the null hypothesis is accepted. if the null hypothesis is accepted. Example: Accepting a shipment of goods from a Example: Accepting a shipment of goods from a supplier or returning the shipment of goods to the supplier or returning the shipment of goods to the supplier supplier

7. 7 7 Slide © 2008 Thomson South-Western. All Rights Reserved One-tailed(lower-tail)One-tailed(upper-tail)Two-tailed Summary of Forms for Null and Alternative Hypotheses about a Population Mean n The equality part of the hypotheses always appears in the null hypothesis. in the null hypothesis. In general, a hypothesis test about the value of a In general, a hypothesis test about the value of a population mean  must take one of the following population mean  must take one of the following three forms (where  0 is the hypothesized value of three forms (where  0 is the hypothesized value of the population mean). the population mean).

8. 8 8 Slide © 2008 Thomson South-Western. All Rights Reserved n Example: Metro EMS Null and Alternative Hypotheses Operating in a multiple Operating in a multiple hospital system with approximately 20 mobile medical units, the service goal is to respond to medical emergencies with a mean time of 12 minutes or less. A major west coast city provides A major west coast city provides one of the most comprehensive emergency medical services in the world.

9. 9 9 Slide © 2008 Thomson South-Western. All Rights Reserved The director of medical services The director of medical services wants to formulate a hypothesis test that could use a sample of emergency response times to determine whether or not the service goal of 12 minutes or less is being achieved. n Example: Metro EMS Null and Alternative Hypotheses

10. 10 Slide © 2008 Thomson South-Western. All Rights Reserved Null and Alternative Hypotheses One-tailed (upper-tail) Test Null - The emergency service is meeting the response goal; no follow-up action is necessary. Alternative - The emergency service is not meeting the response goal; appropriate follow-up action is necessary. H 0 :  H a :  where:  = mean response time for the population of medical emergency requests of medical emergency requests

11. 11 Slide © 2008 Thomson South-Western. All Rights Reserved Type I and Type II Error n In Type I error, the sample mean ( x ) is outside the test limits but the actual population mean (μ ) is inside the test limits. So we think we are wrong, based on the sample mean, but we are actually right. x μ n In Type I error, the sample mean ( x ) is outside the test limits but the actual population mean (μ ) is inside the test limits. So we think we are wrong, based on the sample mean, but we are actually right. x μ n In Type II error, the sample mean is inside the test limits, but the actual population mean is outside the test limits. So we think we are right, based on the sample mean, but we are actually wrong. μ x n In Type II error, the sample mean is inside the test limits, but the actual population mean is outside the test limits. So we think we are right, based on the sample mean, but we are actually wrong. μ x

12. 12 Slide © 2008 Thomson South-Western. All Rights Reserved Type I Error Because hypothesis tests are based on sample data, Because hypothesis tests are based on sample data, we must allow for the possibility of errors. we must allow for the possibility of errors. n A Type I error is rejecting H 0 when it is true. You think you are wrong but you are right. think you are wrong but you are right. n The probability of making a Type I error when the null hypothesis is true as an equality is called the null hypothesis is true as an equality is called the level of significance. level of significance. n Applications of hypothesis testing that only control the Type I error are often called significance tests. the Type I error are often called significance tests.

13. 13 Slide © 2008 Thomson South-Western. All Rights Reserved Type I Error n The Greek symbol a (alpha) is used to denote the level of significance, and common choices for a are.05 and.01. n In practice, the person responsible for the hypothesis test specifies the level of significance. By selecting a, that person is controlling the probability of making a Type I error. n If the cost of making a Type I error is not too high, larger values of a, are typically used. n Applications of hypothesis testing that only control for the Type I error are called significance tests.

14. 14 Slide © 2008 Thomson South-Western. All Rights Reserved Type II Error n Although most applications of hypothesis testing control for the probability of making a Type I error, they do not always control for the probability of making a Type II error. n Therefore, if we decide to accept H 0, we cannot determine how confident we can be with that decision. n Because of the uncertainty associated with making a Type II error when conducting significance tests, statisticians usually recommend that we use the statement “do reject H 0 ” instead of “accept H 0.”

15. 15 Slide © 2008 Thomson South-Western. All Rights Reserved Type II Error n Using the statement “do not reject H 0 ” carries the recommendation to withhold judgment and action. n In effect, by not directly accepting H 0, the statistician avoids the risk of making a Type II error. n In effect, by not directly accepting H 0, the statistician avoids the risk of making a Type II error. n Whenever the probability of making a Type II error has not been determined and controlled, we will not make the statement “accept H 0.” In such situations, only two conclusions are possible: do not reject H 0 or reject H 0.

16. 16 Slide © 2008 Thomson South-Western. All Rights Reserved Type II Error n Although controlling for a Type II error in hypothesis testing is not common, it can be done, as shown later in this chapter. n If proper controls have been established for Type II error, action based on the “accept H 0 ” conclusion can be appropriate.

17. 17 Slide © 2008 Thomson South-Western. All Rights Reserved Type II Error n A Type II error is accepting H 0 when it is false. You think you are right but you are wrong. think you are right but you are wrong. n It is difficult to control for the probability of making a Type II error. a Type II error. n Statisticians avoid the risk of making a Type II error by using “do not reject H 0 ” and not “accept H 0 ”. error by using “do not reject H 0 ” and not “accept H 0 ”. n The possibility of making a Type I or a Type II error is always present in decision making and cannot be is always present in decision making and cannot be eliminated. eliminated.

18. 18 Slide © 2008 Thomson South-Western. All Rights Reserved Type I and Type II Errors CorrectDecision Type II Error CorrectDecision Type I Error Reject H 0 (Conclude  > 12) Accept H 0 (Conclude  < 12) H 0 True (  < 12) H 0 False (  > 12) Conclusion Population Condition

19. 19 Slide © 2008 Thomson South-Western. All Rights Reserved p -Value Approach to One-Tailed Hypothesis Testing A p -value is a probability that provides a measure A p -value is a probability that provides a measure of the evidence against the null hypothesis of the evidence against the null hypothesis provided by the sample. provided by the sample. The smaller the p -value, the more evidence there The smaller the p -value, the more evidence there is against H 0. is against H 0. A small p -value, such as.02, indicates the value of the A small p -value, such as.02, indicates the value of the test statistic is unusual. There are only 2 chances out test statistic is unusual. There are only 2 chances out of 100 that the sample mean can be this small given of 100 that the sample mean can be this small given the assumption that H 0 is true. Such a result is the assumption that H 0 is true. Such a result is unlikely if the null hypothesis is true. unlikely if the null hypothesis is true. The p -value is used to determine if the null The p -value is used to determine if the null hypothesis should be rejected. hypothesis should be rejected.

20. 20 Slide © 2008 Thomson South-Western. All Rights Reserved p -Value Approach to One-Tailed Hypothesis Testing n The level of significance, denoted by a, is the probability of making a Type I error by rejecting H 0 when the null hypothesis is true as an equality. n The p -value is the observed level of significance. n If the cost of making a Type I error is high, a small value should be chosen for the level of significance. If the cost is not high, a larger value is more appropriate.

21. 21 Slide © 2008 Thomson South-Western. All Rights Reserved n p -Value Approach p -value  p -value  0 0 - z  = -1.28 - z  = -1.28  =.10 z z z = -1.46 z = -1.46 Lower-Tailed Test About a Population Mean:  Known Sampling distribution of Sampling distribution of p -Value < , so reject H 0.

22. 22 Slide © 2008 Thomson South-Western. All Rights Reserved n p -Value Approach p -Value  p -Value  0 0 z  = 1.75 z  = 1.75  =.04 z z z = 2.29 z = 2.29 Upper-Tailed Test About a Population Mean:  Known Sampling distribution of Sampling distribution of p -Value < , so reject H 0.

23. 23 Slide © 2008 Thomson South-Western. All Rights Reserved Critical Value Approach to One-Tailed Hypothesis Testing The test statistic z has a standard normal probability The test statistic z has a standard normal probability distribution. distribution. We can use the standard normal probability We can use the standard normal probability distribution table to find the z -value with an area distribution table to find the z -value with an area of  in the lower (or upper) tail of the distribution. of  in the lower (or upper) tail of the distribution. The value of the test statistic that established the The value of the test statistic that established the boundary of the rejection region is called the boundary of the rejection region is called the critical value for the test. critical value for the test. n The rejection rule is: Lower tail: Reject H 0 if z < - z  Lower tail: Reject H 0 if z < - z  Upper tail: Reject H 0 if z > z  Upper tail: Reject H 0 if z > z 

24. 24 Slide © 2008 Thomson South-Western. All Rights Reserved  0 0  z  =  1.28 Reject H 0 Do Not Reject H 0 z Sampling distribution of Sampling distribution of Lower-Tailed Test About a Population Mean:  Known n Critical Value Approach

25. 25 Slide © 2008 Thomson South-Western. All Rights Reserved  0 0 z  = 1.645 Reject H 0 Do Not Reject H 0 z Sampling distribution of Sampling distribution of Upper-Tailed Test About a Population Mean:  Known n Critical Value Approach

26. 26 Slide © 2008 Thomson South-Western. All Rights Reserved Steps of Hypothesis Testing Step 1. Develop the null and alternative hypotheses. Step 2. Specify the level of significance . Step 3. Collect the sample data and compute the test statistic. p -Value Approach Step 4. Use the value of the test statistic to compute the p -value. p -value. Step 5. Reject H 0 if p -value, the observed level of significance, is <  the selected level of significance for the hypothesis test.

27. 27 Slide © 2008 Thomson South-Western. All Rights Reserved Critical Value Approach Step 4. Use the level of significance  to determine the critical value for the test statistic z and the rejection rule. Step 5. Use the value of the test statistic z and the rejection rule to determine whether to reject H 0. Steps of Hypothesis Testing The critical value is the largest value of the test statistic z that will result in the rejection of the null hypothesis.

28. 28 Slide © 2008 Thomson South-Western. All Rights Reserved n Rejection Rule for a Lower Tail Test Reject H 0 if z ≤ - z a where –z a is the critical value that is the z value that provides an area of a in the lower tail of the standard normal distribution. n Rejection Rule for an Upper Tail Test Reject H 0 if z ≥ z a where z a is the critical value that is the z value that provides an area of a in the upper tail of the standard normal distribution. Steps of Hypothesis Testing

29. 29 Slide © 2008 Thomson South-Western. All Rights Reserved n Rejection Rule for a Two Tail Test Reject H 0 if z ≤ - z a or if z ≥ + z a where –z a is the critical value that is the z value that provides an area of a in the lower tail of the standard normal distribution, and z a is the critical value that is the z value that provides an area of a in the upper tail of the standard normal distribution. Steps of Hypothesis Testing

30. 30 Slide © 2008 Thomson South-Western. All Rights Reserved n The p-value approach to hypothesis testing and the critical value approach will always lead to the same rejection decision. That is, for a lower-tail test whenever the p-value is less than or equal to a, the value of the test statistic will be less than or equal to the critical value for the lower tail test. And for an upper-tail test whenever the p-value is less than or equal to a, the value of the test statistic will be greater than or equal to the critical value for the upper tail test. n The advantage of the p-value approach is that the p- value tells us how significant the results are (the observed level of significance). If we use the critical value approach, we only know that the results are significant at the stated level of significance. Steps of Hypothesis Testing

31. 31 Slide © 2008 Thomson South-Western. All Rights Reserved n Example: Metro EMS The EMS director wants to The EMS director wants to perform a hypothesis test, with a.05 level of significance, to determine whether the service goal of 12 minutes or less is being achieved. The response times for a random The response times for a random sample of 40 medical emergencies were tabulated. The sample mean is 13.25 minutes. The population standard deviation is believed to be 3.2 minutes. One-Tailed Tests About a Population Mean:  Known

32. 32 Slide © 2008 Thomson South-Western. All Rights Reserved 1. Develop the hypotheses. 2. Specify the level of significance.  =.05 H 0 :  H a :  p -Value and Critical Value Approaches p -Value and Critical Value Approaches One-Tailed Tests About a Population Mean:  Known 3. Compute the value of the test statistic.

33. 33 Slide © 2008 Thomson South-Western. All Rights Reserved 5. Determine whether to reject H 0. p –Value Approach p –Value Approach One-Tailed Tests About a Population Mean:  Known 4. Compute the p –value. For z = 2.47, cumulative probability =.9932. p –value = 1 .9932 =.0068 Because p –value =.0068 <  =.05, we reject H 0. There is sufficient statistical evidence to infer that Metro EMS is not meeting the response goal of 12 minutes.

34. 34 Slide © 2008 Thomson South-Western. All Rights Reserved n p –Value Approach p -value  p -value  0 0 z  = 1.645 z  = 1.645  =.05 z z z = 2.47 z = 2.47 One-Tailed Tests About a Population Mean:  Known Sampling distribution of Sampling distribution of

35. 35 Slide © 2008 Thomson South-Western. All Rights Reserved 5. Determine whether to reject H 0. There is sufficient statistical evidence to infer that Metro EMS is not meeting the response goal of 12 minutes. Because 2.47 > 1.645, we reject H 0. Critical Value Approach Critical Value Approach One-Tailed Tests About a Population Mean:  Known For  =.05, z.05 = 1.645 4. Determine the critical value and rejection rule. Reject H 0 if z > 1.645

36. 36 Slide © 2008 Thomson South-Western. All Rights Reserved p -Value Approach to Two-Tailed Hypothesis Testing The rejection rule: The rejection rule: Reject H 0 if the p -value < . Reject H 0 if the p -value < . Compute the p -value using the following three steps: Compute the p -value using the following three steps: 3. Double the tail area obtained in step 2 to obtain the p –value. the p –value. 2. If z is in the upper tail ( z > 0), find the area under the standard normal curve to the right of z. the standard normal curve to the right of z. If z is in the lower tail ( z < 0), find the area under If z is in the lower tail ( z < 0), find the area under the standard normal curve to the left of z. the standard normal curve to the left of z. 1. Compute the value of the test statistic z.

37. 37 Slide © 2008 Thomson South-Western. All Rights Reserved Critical Value Approach to Two-Tailed Hypothesis Testing The critical values will occur in both the lower and The critical values will occur in both the lower and upper tails of the standard normal curve. upper tails of the standard normal curve. n The rejection rule is: Reject H 0 if z z  /2. Reject H 0 if z z  /2. Use the standard normal probability distribution Use the standard normal probability distribution table to find z  /2 (the z -value with an area of  /2 in table to find z  /2 (the z -value with an area of  /2 in the upper tail of the distribution). the upper tail of the distribution).

38. 38 Slide © 2008 Thomson South-Western. All Rights Reserved Example: Glow Toothpaste Two-Tailed Test About a Population Mean:  Known Two-Tailed Test About a Population Mean:  Known oz. Glow Quality assurance procedures call for Quality assurance procedures call for the continuation of the filling process if the sample results are consistent with the assumption that the mean filling weight for the population of toothpaste tubes is 6 oz.; otherwise the process will be adjusted. The production line for Glow toothpaste The production line for Glow toothpaste is designed to fill tubes with a mean weight of 6 oz. Periodically, a sample of 30 tubes will be selected in order to check the filling process.

39. 39 Slide © 2008 Thomson South-Western. All Rights Reserved Example: Glow Toothpaste Two-Tailed Test About a Population Mean:  Known Two-Tailed Test About a Population Mean:  Known oz. Glow Perform a hypothesis test, at the.03 Perform a hypothesis test, at the.03 level of significance, to help determine whether the filling process should continue operating or be stopped and corrected. Assume that a sample of 30 toothpaste Assume that a sample of 30 toothpaste tubes provides a sample mean of 6.1 oz. The population standard deviation is believed to be 0.2 oz.

40. 40 Slide © 2008 Thomson South-Western. All Rights Reserved 1. Determine the hypotheses. 2. Specify the level of significance. 3. Compute the value of the test statistic.  =.03 p –Value and Critical Value Approaches p –Value and Critical Value Approaches Glow H 0 :  H a : Two-Tailed Tests About a Population Mean:  Known

41. 41 Slide © 2008 Thomson South-Western. All Rights Reserved Glow Two-Tailed Tests About a Population Mean:  Known 5. Determine whether to reject H 0. p –Value Approach p –Value Approach 4. Compute the p –value. For z = 2.74, cumulative probability =.9969 p –value = 2(1 .9969) =.0062 Because p –value =.0062 <  =.03, we reject H 0. There is sufficient statistical evidence to infer that the alternative hypothesis is true (i.e. the mean filling weight is not 6 ounces). (i.e. the mean filling weight is not 6 ounces).

42. 42 Slide © 2008 Thomson South-Western. All Rights Reserved Glow Two-Tailed Tests About a Population Mean:  Known  /2 =.015  /2 =.015 0 0 z  /2 = 2.17 z z  /2 =.015  /2 =.015 p -Value Approach p -Value Approach - z  /2 = -2.17 z = 2.74 z = -2.74 1/2 p -value =.0031 1/2 p -value =.0031 1/2 p -value =.0031 1/2 p -value =.0031

43. 43 Slide © 2008 Thomson South-Western. All Rights Reserved Critical Value Approach Critical Value Approach Glow Two-Tailed Tests About a Population Mean:  Known 5. Determine whether to reject H 0. There is sufficient statistical evidence to infer that the alternative hypothesis is true (i.e. the mean filling weight is not 6 ounces). (i.e. the mean filling weight is not 6 ounces). Because 2.74 > 2.17, we reject H 0. For  /2 =.03/2 =.015, z.015 = 2.17 4. Determine the critical value and rejection rule. Reject H 0 if z 2.17

44. 44 Slide © 2008 Thomson South-Western. All Rights Reserved  /2 =.015 0 0 2.17 Reject H 0 Do Not Reject H 0 z z Reject H 0 -2.17 Glow Critical Value Approach Critical Value Approach Sampling distribution of Sampling distribution of Two-Tailed Tests About a Population Mean:  Known  /2 =.015

45. 45 Slide © 2008 Thomson South-Western. All Rights Reserved Confidence Interval Approach to Two-Tailed Tests About a Population Mean Select a simple random sample from the population Select a simple random sample from the population and use the value of the sample mean to develop and use the value of the sample mean to develop the confidence interval for the population mean . the confidence interval for the population mean . (Confidence intervals are covered in Chapter 8.) (Confidence intervals are covered in Chapter 8.) If the confidence interval contains the hypothesized If the confidence interval contains the hypothesized value  0, do not reject H 0. Otherwise, reject H 0. value  0, do not reject H 0. Otherwise, reject H 0.

46. 46 Slide © 2008 Thomson South-Western. All Rights Reserved The 97% confidence interval for  is The 97% confidence interval for  is Confidence Interval Approach to Two-Tailed Tests About a Population Mean Glow Because the hypothesized value for the Because the hypothesized value for the population mean,  0 = 6, is not in this interval, the hypothesis-testing conclusion is that the null hypothesis, H 0 :  = 6, can be rejected. or 6.02076 to 6.17924

47. 47 Slide © 2008 Thomson South-Western. All Rights Reserved Guidelines for Interpreting p-values n Less than.01 – overwhelming evidence to conclude H a is true. n Between.01 and.05 – strong evidence to conclude H a is true. n Between.05 and.10 – weak evidence to conclude H a is true. n Greater than.10 – insufficient evidence to conclude H a is true.

48. 48 Slide © 2008 Thomson South-Western. All Rights Reserved n Test Statistic Tests About a Population Mean:  Unknown This test statistic has a t distribution with n - 1 degrees of freedom. with n - 1 degrees of freedom.

49. 49 Slide © 2008 Thomson South-Western. All Rights Reserved n Because the σ unknown case corresponds to situations in which an estimate of the population standard deviation cannot be developed prior to sampling, the sample must be used to develop an estimate of both μ and σ. n Thus, to conduct a hypothesis test about a population mean for the σ unknown case, the sample mean x-bar is used as an estimate of μ and the sample standard deviation s is used as an estimate of σ. Tests About a Population Mean:  Unknown

50. 50 Slide © 2008 Thomson South-Western. All Rights Reserved n Rejection Rule: p -Value Approach H 0 :   Reject H 0 if t > t  Reject H 0 if t < - t  Reject H 0 if t t  H 0 :   H 0 :   Tests About a Population Mean:  Unknown n Rejection Rule: Critical Value Approach Reject H 0 if p –value < 

51. 51 Slide © 2008 Thomson South-Western. All Rights Reserved p -Values and the t Distribution The format of the t distribution table provided in most The format of the t distribution table provided in most statistics textbooks does not have sufficient detail statistics textbooks does not have sufficient detail to determine the exact p -value for a hypothesis test. to determine the exact p -value for a hypothesis test. However, we can still use the t distribution table to However, we can still use the t distribution table to identify a range for the p -value. identify a range for the p -value. An advantage of computer software packages, such An advantage of computer software packages, such as Excel, is that the computer output will provide the as Excel, is that the computer output will provide the p -value for the t distribution. p -value for the t distribution. Before going ahead with use of the t distribution, Before going ahead with use of the t distribution, construct a histogram of the sample data to check on construct a histogram of the sample data to check on the form of the population distribution. You are the form of the population distribution. You are looking for skewness and extreme outliers. looking for skewness and extreme outliers.

52. 52 Slide © 2008 Thomson South-Western. All Rights Reserved Using the t Distribution n When σ is unknown but the population is normally distributed, the hypothesis tests using the t distribution provide exact results for any sample size. n When the population is not normally distributed, the procedures are approximations. n Sample sizes of 30 or greater will provide good results in most cases. If the population is approximately normal, small sample sizes ( n < 15) can provide acceptable results. If the population is highly skewed or contains outliers, sample sizes approaching 50 are recommended.

53. 53 Slide © 2008 Thomson South-Western. All Rights Reserved A State Highway Patrol periodically samples A State Highway Patrol periodically samples vehicle speeds at various locations on a particular roadway. The sample of vehicle speeds is used to test the hypothesis Example: Highway Patrol One-Tailed Test About a Population Mean:  Unknown One-Tailed Test About a Population Mean:  Unknown The locations where H 0 is rejected are deemed The locations where H 0 is rejected are deemed the best locations for radar traps. H 0 :  < 65

54. 54 Slide © 2008 Thomson South-Western. All Rights Reserved Example: Highway Patrol One-Tailed Test About a Population Mean:  Unknown One-Tailed Test About a Population Mean:  Unknown At Location F, a sample of 64 vehicles shows a At Location F, a sample of 64 vehicles shows a mean speed of 66.2 mph with a standard deviation of 4.2 mph. Use  =.05 to test the hypothesis.

55. 55 Slide © 2008 Thomson South-Western. All Rights Reserved One-Tailed Test About a Population Mean:  Unknown 1. Determine the hypotheses. 2. Specify the level of significance. 3. Compute the value of the test statistic.  =.05 p –Value and Critical Value Approaches p –Value and Critical Value Approaches H 0 :  < 65 H a :  > 65

56. 56 Slide © 2008 Thomson South-Western. All Rights Reserved One-Tailed Test About a Population Mean:  Unknown p –Value Approach p –Value Approach 5. Determine whether to reject H 0. 4. Compute the p –value. For t = 2.286, the p –value must be less than.025 (for t = 1.998) and greater than.01 (for t = 2.387)..01 < p –value <.025 Because p –value <  =.05, we reject H 0. We are at least 95% confident that the mean speed of vehicles at Location F is greater than 65 mph. of vehicles at Location F is greater than 65 mph.

57. 57 Slide © 2008 Thomson South-Western. All Rights Reserved Critical Value Approach Critical Value Approach 5. Determine whether to reject H 0. We are at least 95% confident that the mean speed of vehicles at Location F is greater than 65 mph. Location F is a good candidate for a radar trap. Because 2.286 > 1.669, we reject H 0. One-Tailed Test About a Population Mean:  Unknown For  =.05 and d.f. = 64 – 1 = 63, t.05 = 1.669 4. Determine the critical value and rejection rule. Reject H 0 if t > 1.669

58. 58 Slide © 2008 Thomson South-Western. All Rights Reserved  0 0 t  = 1.669 t  = 1.669 Reject H 0 Do Not Reject H 0 t One-Tailed Test About a Population Mean:  Unknown

59. 59 Slide © 2008 Thomson South-Western. All Rights Reserved End of Chapter 9, Part A