1. 1 1 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. John Loucks St. Edward’s University...................... SLIDES. BY

2. 2 2 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Hypothesis Testing 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. should or should not be rejected. 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 hypothesis testing procedure uses data from a The hypothesis testing procedure uses data from a sample to test the two competing statements sample to test the two competing statements indicated by H 0 and H a. indicated by H 0 and H a.

3. 3 3 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Developing Null and Alternative Hypotheses It is not always obvious how the null and alternative It is not always obvious how the null and alternative hypotheses should be formulated. hypotheses should be formulated. Care must be taken to structure the hypotheses Care must be taken to structure the hypotheses appropriately so that the test conclusion provides appropriately so that the test conclusion provides the information the researcher wants. the information the researcher wants. The context of the situation is very important in The context of the situation is very important in determining how the hypotheses should be stated. determining how the hypotheses should be stated. In some cases it is easier to identify the alternative In some cases it is easier to identify the alternative hypothesis first. In other cases the null is easier. hypothesis first. In other cases the null is easier. Correct hypothesis formulation will take practice. Correct hypothesis formulation will take practice.

4. 4 4 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Alternative Hypothesis as a Research Hypothesis Developing Null and Alternative Hypotheses Many applications of hypothesis testing involve Many applications of hypothesis testing involve an attempt to gather evidence in support of a an attempt to gather evidence in support of a research hypothesis. research hypothesis. In such cases, it is often best to begin with the In such cases, it is often best to begin with the alternative hypothesis and make it the conclusion alternative hypothesis and make it the conclusion that the researcher hopes to support. that the researcher hopes to support. The conclusion that the research hypothesis is true The conclusion that the research hypothesis is true is made if the sample data provide sufficient is made if the sample data provide sufficient evidence to show that the null hypothesis can be evidence to show that the null hypothesis can be rejected. rejected.

5. 5 5 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Alternative Hypothesis as a Research Hypothesis Developing Null and Alternative Hypotheses Example: Example: A new teaching method is developed that is A new teaching method is developed that is believed to be better than the current method. believed to be better than the current method. Alternative Hypothesis: Alternative Hypothesis: The new teaching method is better. The new teaching method is better. Null Hypothesis: Null Hypothesis: The new method is no better than the old method. The new method is no better than the old method.

6. 6 6 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Alternative Hypothesis as a Research Hypothesis Developing Null and Alternative Hypotheses Example: Example: A new sales force bonus plan is developed in an A new sales force bonus plan is developed in an attempt to increase sales. attempt to increase sales. Alternative Hypothesis: Alternative Hypothesis: The new bonus plan increase sales. The new bonus plan increase sales. Null Hypothesis: Null Hypothesis: The new bonus plan does not increase sales. The new bonus plan does not increase sales.

7. 7 7 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Alternative Hypothesis as a Research Hypothesis Developing Null and Alternative Hypotheses Example: Example: A new drug is developed with the goal of lowering A new drug is developed with the goal of lowering blood pressure more than the existing drug. blood pressure more than the existing drug. Alternative Hypothesis: Alternative Hypothesis: The new drug lowers blood pressure more than The new drug lowers blood pressure more than the existing drug. the existing drug. Null Hypothesis: Null Hypothesis: The new drug does not lower blood pressure more The new drug does not lower blood pressure more than the existing drug. than the existing drug.

8. 8 8 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Developing Null and Alternative Hypotheses n Null Hypothesis as an Assumption to be Challenged We might begin with a belief or assumption that We might begin with a belief or assumption that a statement about the value of a population a statement about the value of a population parameter is true. parameter is true. We then using a hypothesis test to challenge the We then using a hypothesis test to challenge the assumption and determine if there is statistical assumption and determine if there is statistical evidence to conclude that the assumption is evidence to conclude that the assumption is incorrect. incorrect. In these situations, it is helpful to develop the null In these situations, it is helpful to develop the null hypothesis first. hypothesis first.

9. 9 9 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Developing Null and Alternative Hypotheses Example: Example: The label on a soft drink bottle states that it The label on a soft drink bottle states that it contains 67.6 fluid ounces. contains 67.6 fluid ounces. Null Hypothesis: Null Hypothesis: The label is correct.  > 67.6 ounces. The label is correct.  > 67.6 ounces. Alternative Hypothesis: Alternative Hypothesis: The label is incorrect.  < 67.6 ounces. The label is incorrect.  < 67.6 ounces. n Null Hypothesis as an Assumption to be Challenged

10. 10 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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).

11. 11 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Example: Metro EMS Null and Alternative Hypotheses A major west coast city provides one of the most 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 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.

12. 12 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Null and Alternative Hypotheses The emergency service is meeting the response goal; no follow-up action is necessary. 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

13. 13 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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. 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 p-value. p-value. n The maximum p-value that can be tolerated and still reject a null hypothesis that could be true is still reject a null hypothesis that could be true is called the level of significance . called the level of significance .

14. 14 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Type II Error n A Type II error is accepting H 0 when it is false. 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 ”.

15. 15 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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

16. 16 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. p -Value Approach to One-Tailed Hypothesis Testing Reject H 0 if the p -value < . Reject H 0 if the p -value < . The p -value is the probability, computed using the The p -value is the probability, computed using the test statistic, that measures the support (or lack of test statistic, that measures the support (or lack of support) provided by the sample for the null support) provided by the sample for the null hypothesis. hypothesis. If the p -value is less than or equal to the level of If the p -value is less than or equal to the level of significance , the value of the test statistic is in the significance , the value of the test statistic is in the rejection region. rejection region.

17. 17 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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 < .

18. 18 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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.

19. 19 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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 < 

20. 20 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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 is that An advantage of computer software packages is that the computer output will provide the p -value for the the computer output will provide the p -value for the t distribution. t distribution.

21. 21 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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 H 0 :  < 65. 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 The locations where H 0 is rejected are deemed the best locations for radar traps. 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.

22. 22 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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

23. 23 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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.

24. 24 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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

25. 25 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part.  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

26. 26 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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 proportion p must take one of the population proportion p must take one of the following three forms (where p 0 is the hypothesized following three forms (where p 0 is the hypothesized value of the population proportion). value of the population proportion). A Summary of Forms for Null and Alternative Hypotheses About a Population Proportion One-tailed (lower tail) One-tailed (upper tail) Two-tailed H 0 : p > p 0 H a : p < p 0 H 0 : p < p 0 H a : p > p 0 H 0 : p = p 0 H a : p ≠ p 0

27. 27 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Test Statistic Tests About a Population Proportion where: assuming np > 5 and n (1 – p ) > 5

28. 28 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Rejection Rule: p –Value Approach H 0 : p  p  Reject H 0 if z > z  Reject H 0 if z < - z  Reject H 0 if z z  H 0 : p  p  H 0 : p  p  Tests About a Population Proportion Reject H 0 if p –value <  n Rejection Rule: Critical Value Approach

29. 29 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Example: National Safety Council (NSC) For a Christmas and New Year’s week, the 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. Two-Tailed Test About a Population Proportion A sample of 120 accidents showed that 67 were A sample of 120 accidents showed that 67 were caused by drunk driving. Use these data to test the NSC’s claim with  =.05.

30. 30 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Two-Tailed Test About a Population Proportion 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 a common error is using in this formula in this formula

31. 31 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. p  Value Approach p  Value Approach 4. Compute the p -value. 5. Determine whether to reject H 0. Because p –value =.2006 >  =.05, we cannot reject H 0. Two-Tailed Test About a Population Proportion For z = 1.28, cumulative probability =.8997 p –value = 2(1 .8997) =.2006

32. 32 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Two-Tailed Test About a Population Proportion Critical Value Approach Critical Value Approach 5. Determine whether to reject H 0. For  /2 =.05/2 =.025, z.025 = 1.96 4. Determine the criticals value and rejection rule. Reject H 0 if z 1.96 Because 1.278 > -1.96 and -1.96 and < 1.96, we cannot reject H 0.

33. 33 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Hypothesis Testing and Decision Making n Thus far, we have illustrated hypothesis testing applications referred to as significance tests. applications referred to as significance tests. n With a significance test, we control the probability of making the Type I error, but not the Type II error. making the Type I error, but not the Type II error. n In the tests, we compared the p -value to a controlled probability of a Type I error, , which is called the probability of a Type I error, , which is called the level of significance for the test. level of significance for the test. n We recommended the conclusion “do not reject H 0 ” rather than “accept H 0 ” because the latter puts us at rather than “accept H 0 ” because the latter puts us at risk of making a Type II error. risk of making a Type II error.

34. 34 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Hypothesis Testing and Decision Making n In many decision-making situations the decision maker may want, and in some cases may be forced, maker may want, and in some cases may be forced, to take action with both the conclusion “do not reject to take action with both the conclusion “do not reject H 0 “and the conclusion “reject H 0.” H 0 “and the conclusion “reject H 0.” n In such situations, it is recommended that the hypothesis-testing procedure be extended to include hypothesis-testing procedure be extended to include consideration of making a Type II error. consideration of making a Type II error. n With the conclusion “do not reject H 0 ”, the statistical evidence is considered inconclusive. evidence is considered inconclusive. n Usually this is an indication to postpone a decision until further research and testing is undertaken. until further research and testing is undertaken.

35. 35 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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.