1 1 Slide © 2003 South-Western/Thomson Learning™ Slides Prepared by JOHN S. LOUCKS St. Edward’s University

2 2 Slide © 2003 South-Western/Thomson Learning™ Chapter 9 Hypothesis Testing n Developing Null and Alternative Hypotheses n Type I and Type II Errors n One-Tailed Tests About a Population Mean: Large-Sample Case Large-Sample Case n Two-Tailed Tests About a Population Mean: Large-Sample Case Large-Sample Case n Tests About a Population Mean: Small-Sample Case continued

3 3 Slide © 2003 South-Western/Thomson Learning™ Chapter 9 Hypothesis Testing n Tests About a Population Proportion n Hypothesis Testing and Decision Making n Calculating the Probability of Type II Errors n Determining the Sample Size for a Hypothesis Test about a Population Mean about a Population Mean

4 4 Slide © 2003 South-Western/Thomson Learning™ Developing Null and Alternative Hypotheses n Hypothesis testing can be used to determine whether a statement about the value of a population parameter should or should not be rejected. n The null hypothesis, denoted by H 0, is a tentative assumption about a population parameter. n The alternative hypothesis, denoted by H a, is the opposite of what is stated in the null hypothesis.

5 5 Slide © 2003 South-Western/Thomson Learning™ n Testing Research Hypotheses The research hypothesis should be expressed as the alternative hypothesis. The research hypothesis should be expressed as the alternative hypothesis. The conclusion that the research hypothesis is true comes from sample data that contradict the null hypothesis. The conclusion that the research hypothesis is true comes from sample data that contradict the null hypothesis. Developing Null and Alternative Hypotheses

6 6 Slide © 2003 South-Western/Thomson Learning™ Developing Null and Alternative Hypotheses n Testing the Validity of a Claim Manufacturers’ claims are usually given the benefit of the doubt and stated as the null hypothesis. Manufacturers’ claims are usually given the benefit of the doubt and stated as the null hypothesis. The conclusion that the claim is false comes from sample data that contradict the null hypothesis. The conclusion that the claim is false comes from sample data that contradict the null hypothesis.

7 7 Slide © 2003 South-Western/Thomson Learning™ n Testing in Decision-Making Situations A decision maker might have to choose between two courses of action, one associated with the null hypothesis and another associated with the alternative hypothesis. A decision maker might have to choose between two courses of action, one associated with the null hypothesis and another associated with the alternative hypothesis. Example: Accepting a shipment of goods from a supplier or returning the shipment of goods to the supplier. Example: Accepting a shipment of goods from a supplier or returning the shipment of goods to the supplier. Developing Null and Alternative Hypotheses

8 8 Slide © 2003 South-Western/Thomson Learning™ 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 general, a hypothesis test about the value of a population mean  must take one of the following three forms (where  0 is the hypothesized value of the population mean). In general, a hypothesis test about the value of a population mean  must take one of the following three forms (where  0 is the hypothesized value of the population mean). H 0 :  >  0 H 0 :   0 H 0 :  <  0 H 0 :  =  0 H a :   0 H a :   0 H a :   0 H a :   0

9 9 Slide © 2003 South-Western/Thomson Learning™ Example: Metro EMS n Null and Alternative Hypotheses A major west coast city provides one of the most comprehensive emergency medical services in the world. 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. 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.

10 Slide © 2003 South-Western/Thomson Learning™ Example: Metro EMS n Null and Alternative Hypotheses Hypotheses Conclusion and Action H 0 :  The emergency service is meeting the response goal; no follow-up the response goal; no follow-up action is necessary. action is necessary. H a :  The emergency service is not H a :  The emergency service is not meeting the response goal; meeting the response goal; appropriate follow-up action is appropriate follow-up action is necessary. necessary. Where:  = mean response time for the population of medical emergency requests. of medical emergency requests.

11 Slide © 2003 South-Western/Thomson Learning™ Type I Errors n Since hypothesis tests are based on sample data, we must allow for the possibility of errors. n A Type I error is rejecting H 0 when it is true. n The person conducting the hypothesis test specifies the maximum allowable probability of making a Type I error, denoted by  and called the level of significance.

12 Slide © 2003 South-Western/Thomson Learning™ Type II Errors n A Type II error is accepting H 0 when it is false. Generally, we cannot control for the probability of making a Type II error, denoted by . Generally, we cannot control for the probability of making a Type II error, denoted by . n Statisticians avoid the risk of making a Type II error by using “do not reject H 0 ” and not “accept H 0 ”.

13 Slide © 2003 South-Western/Thomson Learning™ n Type I and Type II Errors Population Condition Population Condition H 0 True H a True H 0 True H a True Conclusion (  ) (  ) Accept H 0 Correct Type II Accept H 0 Correct Type II (conclude  Conclusion Error (conclude  Conclusion Error Reject H 0 Type I Correct Reject H 0 Type I Correct (conclude  rror Conclusion (conclude  rror Conclusion Example: Metro EMS

14 Slide © 2003 South-Western/Thomson Learning™ The Use of p -Values n The p -value is the probability of obtaining a sample result that is at least as unlikely as what is observed. n The p -value can be used to make the decision in a hypothesis test by noting that: if the p -value is less than the level of significance , the value of the test statistic is in the rejection region. if the p -value is less than the level of significance , the value of the test statistic is in the rejection region. if the p -value is greater than or equal to , the value of the test statistic is not in the rejection region. if the p -value is greater than or equal to , the value of the test statistic is not in the rejection region. Reject H 0 if the p -value < . Reject H 0 if the p -value < .

15 Slide © 2003 South-Western/Thomson Learning™ Steps of Hypothesis Testing 1.Develop the null and alternative hypotheses. 2.Specify the level of significance . 3.Select the test statistic that will be used to test the hypothesis. Using the Test Statistic: 4.Use  to determine the critical value(s) for the test statistic and state the rejection rule for H 0. 5.Collect the sample data and compute the value of the test statistic. 6.Use the value of the test statistic and the rejection rule to determine whether to reject H 0. continued

16 Slide © 2003 South-Western/Thomson Learning™ Steps of Hypothesis Testing Using the p -Value: 4.Collect the sample data and compute the value of the test statistic. 5.Use the value of the test statistic to compute the p - value. 6.Reject H 0 if p -value < .

17 Slide © 2003 South-Western/Thomson Learning™ n Hypotheses Left-Tailed Right-Tailed H 0 :   H 0 :   H a :   H a :   Test Statistic  Known  Unknown Test Statistic  Known  Unknown n Rejection Rule Left-Tailed Right-Tailed Reject H 0 if z > z  Reject H 0 if z z  Reject H 0 if z < - z  One-Tailed Tests about a Population Mean: Large-Sample Case ( n > 30)

18 Slide © 2003 South-Western/Thomson Learning™ Example: Metro EMS n One-Tailed Test about a Population Mean: Large n Let  = P (Type I Error) =.05 Sampling distribution of (assuming H 0 is true and  = 12) Sampling distribution of (assuming H 0 is true and  = 12)  12 c c Reject H 0 Do Not Reject H  (Critical value)

19 Slide © 2003 South-Western/Thomson Learning™ Example: Metro EMS n One-Tailed Test about a Population Mean: Large n Let n = 40, = minutes, s = 3.2 minutes Let n = 40, = minutes, s = 3.2 minutes (The sample standard deviation s can be used to (The sample standard deviation s can be used to estimate the population standard deviation .) estimate the population standard deviation .) Since 2.47 > 1.645, we reject H 0. Since 2.47 > 1.645, we reject H 0. Conclusion : We are 95% confident that Metro EMS Conclusion : We are 95% confident that Metro EMS is not meeting the response goal of 12 minutes; is not meeting the response goal of 12 minutes; appropriate action should be taken to improve appropriate action should be taken to improve service. service.

20 Slide © 2003 South-Western/Thomson Learning™ Example: Metro EMS n Using the p -value to Test the Hypothesis Recall that z = 2.47 for = Then p -value = Since p -value < , that is.0068 <.05, we reject H 0. n Using the p -value to Test the Hypothesis Recall that z = 2.47 for = Then p -value = Since p -value < , that is.0068 <.05, we reject H 0. p -value  Do Not Reject H 0 Reject H 0 z z 2.47

21 Slide © 2003 South-Western/Thomson Learning™ Using Excel to Conduct a One-Tailed Hypothesis Test n Formula Worksheet Note: Rows are not shown.

22 Slide © 2003 South-Western/Thomson Learning™ Using Excel to Conduct a One-Tailed Hypothesis Test n Value Worksheet Note: Rows are not shown.

23 Slide © 2003 South-Western/Thomson Learning™ n Hypotheses H 0 :   H a :   Test Statistic  Known  Unknown Test Statistic  Known  Unknown n Rejection Rule Reject H 0 if | z | > z  Reject H 0 if | z | > z  Two-Tailed Tests about a Population Mean: Large-Sample Case ( n > 30)

24 Slide © 2003 South-Western/Thomson Learning™ Example: Glow Toothpaste n Two-Tailed Tests about a Population Mean: Large n The production line for Glow toothpaste is designed to fill tubes of toothpaste with a mean weight of 6 ounces. Periodically, a sample of 30 tubes will be selected in order to check the filling process. 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 ounces; otherwise the filling process will be stopped and adjusted.

25 Slide © 2003 South-Western/Thomson Learning™ Example: Glow Toothpaste n Two-Tailed Tests about a Population Mean: Large n A hypothesis test about the population mean can be used to help determine when the filling process should continue operating and when it should be stopped and corrected. Hypotheses Hypotheses H 0 :   H 0 :    H a :  Rejection Rule Rejection Rule  ssuming a.05 level of significance, Reject H 0 if z 1.96 Reject H 0 if z 1.96

26 Slide © 2003 South-Western/Thomson Learning™ Example: Glow Toothpaste n Two-Tailed Test about a Population Mean: Large n  Sampling distribution of (assuming H 0 is true and  = 6) Sampling distribution of (assuming H 0 is true and  = 6) Reject H 0 Do Not Reject H 0 z z Reject H 

27 Slide © 2003 South-Western/Thomson Learning™ Example: Glow Toothpaste n Two-Tailed Test about a Population Mean: Large n Assume that a sample of 30 toothpaste tubes provides a sample mean of 6.1 ounces and standard deviation of 0.2 ounces. Let n = 30, = 6.1 ounces, s =.2 ounces n Two-Tailed Test about a Population Mean: Large n Assume that a sample of 30 toothpaste tubes provides a sample mean of 6.1 ounces and standard deviation of 0.2 ounces. Let n = 30, = 6.1 ounces, s =.2 ounces

28 Slide © 2003 South-Western/Thomson Learning™ Example: Glow Toothpaste Conclusion : Since 2.74 > 1.96, we reject H 0. We are 95% confident that the mean filling weight of the toothpaste tubes is not 6 ounces. The filling process should be examined and most likely adjusted.

29 Slide © 2003 South-Western/Thomson Learning™ Example: Glow Toothpaste n Using the p -Value for a Two-Tailed Hypothesis Test Suppose we define the p -value for a two-tailed test as double the area found in the tail of the distribution. With z = 2.74, the standard normal probability table shows there is a =.0031 probability of a difference larger than.1 in the upper tail of the distribution. Considering the same probability of a larger difference in the lower tail of the distribution, we have p -value = 2(.0031) =.0062 p -value = 2(.0031) =.0062 The p -value.0062 is less than  =.05, so H 0 is rejected.

30 Slide © 2003 South-Western/Thomson Learning™ Using Excel to Conduct a Two-Tailed Hypothesis Test n Formula Worksheet Note: Rows are not shown.

31 Slide © 2003 South-Western/Thomson Learning™ n Value Worksheet Using Excel to Conduct a Two-Tailed Hypothesis Test Note: Rows are not shown.

32 Slide © 2003 South-Western/Thomson Learning™ Confidence Interval Approach to a Two-Tailed Test about a Population Mean Select a simple random sample from the population and use the value of the sample mean to develop the confidence interval for the population mean . Select a simple random sample from the population and use the value of the sample mean to develop the confidence interval for the population mean . If the confidence interval contains the hypothesized value  0, do not reject H 0. Otherwise, reject H 0. If the confidence interval contains the hypothesized value  0, do not reject H 0. Otherwise, reject H 0.

33 Slide © 2003 South-Western/Thomson Learning™ Example: Glow Toothpaste n Confidence Interval Approach to a Two-Tailed Hypothesis Test The 95% confidence interval for  is The 95% confidence interval for  is or to or to Since 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.

34 Slide © 2003 South-Western/Thomson Learning™ Test Statistic  Known  Unknown Test Statistic  Known  Unknown This test statistic has a t distribution with n - 1 degrees of freedom. n Rejection Rule One-Tailed Two-Tailed One-Tailed Two-Tailed H 0 :   Reject H 0 if t > t  H 0 :   Reject H 0 if t > t  H 0 :   Reject H 0 if t < - t  H 0 :   Reject H 0 if t < - t  H 0 :   Reject H 0 if | t | > t  H 0 :   Reject H 0 if | t | > t  Tests about a Population Mean: Small-Sample Case ( n < 30)

35 Slide © 2003 South-Western/Thomson Learning™ p -Values and the t Distribution n The format of the t distribution table provided in most statistics textbooks does not have sufficient detail to determine the exact p -value for a hypothesis test. n However, we can still use the t distribution table to identify a range for the p -value. n An advantage of computer software packages is that the computer output will provide the p -value for the t distribution.

36 Slide © 2003 South-Western/Thomson Learning™ Example: Highway Patrol n One-Tailed Test about a Population Mean: Small n 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 H 0 :  < 65. H 0 :  < 65. The locations where H 0 is rejected are deemed the best locations for radar traps. At Location F, a sample of 16 vehicles shows a mean speed of 68.2 mph with a standard deviation of 3.8 mph. Use an  =.05 to test the hypothesis.

37 Slide © 2003 South-Western/Thomson Learning™ Example: Highway Patrol n One-Tailed Test about a Population Mean: Small n Let n = 16, = 68.2 mph, s = 3.8 mph Let n = 16, = 68.2 mph, s = 3.8 mph  =.05, d.f. = 16-1 = 15, t  =  =.05, d.f. = 16-1 = 15, t  = Since 3.37 > 1.753, we reject H 0. Since 3.37 > 1.753, we reject H 0. Conclusion : We are 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.

38 Slide © 2003 South-Western/Thomson Learning™ Using Excel to Conduct a One-Tailed Hypothesis Test: Small-Sample Case n Formula Worksheet Note: Rows are not shown.

39 Slide © 2003 South-Western/Thomson Learning™ n Value Worksheet Using Excel to Conduct a One-Tailed Hypothesis Test: Small-Sample Case Note: Rows are not shown.

40 Slide © 2003 South-Western/Thomson Learning™ Summary of Test Statistics to be Used in a Hypothesis Test about a Population Mean n > 30 ? assumedknown ? Population approximately normal normal ? Use s to estimate  Use s to estimate  Use s to estimate  Increase n to > 30 Yes Yes Yes Yes No No No No assumedknown ?

41 Slide © 2003 South-Western/Thomson Learning™ Summary of Forms for Null and Alternative Hypotheses about a Population Proportion n The equality part of the hypotheses always appears in the null hypothesis. n In general, a hypothesis test about the value of a population proportion p must take one of the following three forms (where p 0 is the hypothesized value of the population proportion). H 0 : p > p 0 H 0 : p p 0 H 0 : p < p 0 H 0 : p = p 0 H a : p p 0 H a : p p 0 H a : p p 0 H a : p p 0

42 Slide © 2003 South-Western/Thomson Learning™ Tests about a Population Proportion: Large-Sample Case ( np > 5 and n (1 - p ) > 5) n Test Statistic where: n Rejection Rule One-Tailed Two-Tailed One-Tailed Two-Tailed H 0 : p  p  Reject H 0 if z > z  H 0 : p  p  Reject H 0 if z > z  H 0 : p  p  Reject H 0 if z < -z  H 0 : p  p  Reject H 0 if z < -z  H 0 : p  p  Reject H 0 if |z| > z  H 0 : p  p  Reject H 0 if |z| > z 

43 Slide © 2003 South-Western/Thomson Learning™ Example: NSC n Two-Tailed Test about a Population Proportion: Large n For a Christmas and New Year’s week, the National Safety Council estimated that 500 people would be killed and 25,000 injured on the nation’s roads. The NSC claimed that 50% of the accidents would be caused by drunk driving. A sample of 120 accidents showed that 67 were caused by drunk driving. Use these data to test the NSC’s claim with  = 0.05.

44 Slide © 2003 South-Western/Thomson Learning™ Example: NSC n Two-Tailed Test about a Population Proportion: Large n Hypothesis Hypothesis H 0 : p =.5 H 0 : p =.5 H a : p.5 H a : p.5 Test Statistic Test Statistic

45 Slide © 2003 South-Western/Thomson Learning™ Example: NSC n Two-Tailed Test about a Population Proportion: Large n Rejection Rule Rejection Rule Reject H 0 if z 1.96 Conclusion Conclusion Do not reject H 0. For z = 1.278, the p -value is.201. If we reject H 0, we exceed the maximum allowed risk of committing a Type I error ( p -value >.050).

46 Slide © 2003 South-Western/Thomson Learning™ n Formula Worksheet Using Excel to Conduct Hypothesis Tests about a Population Proportion Note: Rows are not shown.

47 Slide © 2003 South-Western/Thomson Learning™ Using Excel to Conduct Hypothesis Tests about a Population Proportion n Value Worksheet Note: Rows are not shown.

48 Slide © 2003 South-Western/Thomson Learning™ Hypothesis Testing and Decision Making n In many decision-making situations the decision maker may want, and in some cases may be forced, to take action with both the conclusion do not reject H 0 and the conclusion reject H 0. n In such situations, it is recommended that the hypothesis-testing procedure be extended to include consideration of making a Type II error.

49 Slide © 2003 South-Western/Thomson Learning™ Calculating the Probability of a Type II Error in Hypothesis Tests about a Population Mean 1. Formulate the null and alternative hypotheses. 2. Use the level of significance  to establish a rejection rule based on the test statistic. 3. Using the rejection rule, solve for the value of the sample mean that identifies the rejection region. 4. Use the results from step 3 to state the values of the sample mean that lead to the acceptance of H 0 ; it also defines the acceptance region. 5. Using the sampling distribution of for any value of  from the alternative hypothesis, and the acceptance region from step 4, compute the probability that the sample mean will be in the acceptance region.

50 Slide © 2003 South-Western/Thomson Learning™ Example: Metro EMS (revisited) n Calculating the Probability of a Type II Error 1. Hypotheses are: H 0 :  and H a :  2. Rejection rule is: Reject H 0 if z > Value of the sample mean that identifies the rejection region: 4. We will accept H 0 when x <

51 Slide © 2003 South-Western/Thomson Learning™ Example: Metro EMS (revisited) n Calculating the Probability of a Type II Error 5. Probabilities that the sample mean will be in the acceptance region: Values of   1-  Values of   1- 

52 Slide © 2003 South-Western/Thomson Learning™ Example: Metro EMS (revisited) n Calculating the Probability of a Type II Error Observations about the preceding table: When the true population mean  is close to the null hypothesis value of 12, there is a high probability that we will make a Type II error. When the true population mean  is close to the null hypothesis value of 12, there is a high probability that we will make a Type II error. When the true population mean  is far above the null hypothesis value of 12, there is a low probability that we will make a Type II error. When the true population mean  is far above the null hypothesis value of 12, there is a low probability that we will make a Type II error.

53 Slide © 2003 South-Western/Thomson Learning™ Power of the Test n The probability of correctly rejecting H 0 when it is false is called the power of the test. For any particular value of , the power is 1 – . For any particular value of , the power is 1 – . We can show graphically the power associated with each value of  ; such a graph is called a power curve. We can show graphically the power associated with each value of  ; such a graph is called a power curve.

54 Slide © 2003 South-Western/Thomson Learning™ where z  = z value providing an area of  in the tail z  = z value providing an area of  in the tail  = population standard deviation  0 = value of the population mean in H 0  a = value of the population mean used for the Type II error Note: In a two-tailed hypothesis test, use z  /2 not z  Determining the Sample Size for a Hypothesis Test About a Population Mean

55 Slide © 2003 South-Western/Thomson Learning™ Relationship among , , and n n Once two of the three values are known, the other can be computed. For a given level of significance , increasing the sample size n will reduce . For a given level of significance , increasing the sample size n will reduce . For a given sample size n, decreasing  will increase , whereas increasing  will decrease b. For a given sample size n, decreasing  will increase , whereas increasing  will decrease b.

56 Slide © 2003 South-Western/Thomson Learning™ End of Chapter 9