1. McGraw-Hill/Irwin Copyright © 2009 by The McGraw-Hill Companies, Inc. One-Sample Hypothesis Testing Chapter99 Logic of Hypothesis Testing Statistical Hypothesis Testing Testing a Mean: Known Population Variance Testing a Mean: Unknown Population Variance Testing a Proportion Power Curves and OC Curves (Optional) Tests for One Variance (Optional)

2. 9-2 Logic of Hypothesis Testing Steps in Hypothesis TestingSteps in Hypothesis Testing Step 1: State the assumption to be tested Step 2: Specify the decision rule Step 3: Collect the data to test the hypothesis Step 4: Make a decision Step 5: Take action based on the decision

3. 9-3 Logic of Hypothesis Testing Hypotheses are a pair of mutually exclusive, collectively exhaustive statements about the world.Hypotheses are a pair of mutually exclusive, collectively exhaustive statements about the world. One statement or the other must be true, but they cannot both be true.One statement or the other must be true, but they cannot both be true. H 0 : Null Hypothesis H 1 : Alternative HypothesisH 0 : Null Hypothesis H 1 : Alternative Hypothesis These two statements are hypotheses because the truth is unknown.These two statements are hypotheses because the truth is unknown. State the Hypothesis State the Hypothesis

4. 9-4 Logic of Hypothesis Testing Efforts will be made to reject the null hypothesis.Efforts will be made to reject the null hypothesis. If H 0 is rejected, we tentatively conclude H 1 to be the case.If H 0 is rejected, we tentatively conclude H 1 to be the case. H 0 is sometimes called the maintained hypothesis.H 0 is sometimes called the maintained hypothesis. H 1 is called the action alternative because action may be required if we reject H 0 in favor of H 1.H 1 is called the action alternative because action may be required if we reject H 0 in favor of H 1. State the Hypothesis State the Hypothesis

5. 9-5 Logic of Hypothesis Testing We cannot prove a null hypothesis, we can only fail to reject it.We cannot prove a null hypothesis, we can only fail to reject it. Can Hypotheses be Proved? Can Hypotheses be Proved? The null hypothesis is assumed true and a contradiction is sought.The null hypothesis is assumed true and a contradiction is sought. Role of Evidence Role of Evidence

6. 9-6 Logic of Hypothesis Testing Types of Error Types of Error Type I error: Rejecting the null hypothesis when it is true. This occurs with probability .Type I error: Rejecting the null hypothesis when it is true. This occurs with probability . Type II error: Failure to reject the null hypothesis when it is false. This occurs with probability .Type II error: Failure to reject the null hypothesis when it is false. This occurs with probability .

7. 9-7 Statistical Hypothesis Testing A statistical hypothesis is a statement about the value of a population parameter .A statistical hypothesis is a statement about the value of a population parameter . A hypothesis test is a decision between two competing mutually exclusive and collectively exhaustive hypotheses about the value of .A hypothesis test is a decision between two competing mutually exclusive and collectively exhaustive hypotheses about the value of . Left-Tailed Test Right-Tailed Test Two-Tailed Test

8. 9-8 Statistical Hypothesis Testing The direction of the test is indicated by H 1 :The direction of the test is indicated by H 1 : > indicates a right-tailed test < indicates a left-tailed test ≠ indicates a two-tailed test

9. 9-9 Statistical Hypothesis Testing When to use a One- or Two-Sided Test When to use a One- or Two-Sided Test A two-sided hypothesis test (i.e.,  ≠  0 ) is used when direction ( ) is of no interest to the decision makerA two-sided hypothesis test (i.e.,  ≠  0 ) is used when direction ( ) is of no interest to the decision maker A one-sided hypothesis test is used when - the consequences of rejecting H 0 are asymmetric, or - where one tail of the distribution is of special importance to the researcher.A one-sided hypothesis test is used when - the consequences of rejecting H 0 are asymmetric, or - where one tail of the distribution is of special importance to the researcher. Rejection in a two-sided test guarantees rejection in a one-sided test, other things being equal.Rejection in a two-sided test guarantees rejection in a one-sided test, other things being equal.

10. 9-10 Statistical Hypothesis Testing Decision Rule Decision Rule A test statistic shows how far the sample estimate is from its expected value, in terms of its own standard error.A test statistic shows how far the sample estimate is from its expected value, in terms of its own standard error. The decision rule uses the known sampling distribution of the test statistic to establish the critical value that divides the sampling distribution into two regions.The decision rule uses the known sampling distribution of the test statistic to establish the critical value that divides the sampling distribution into two regions. Reject H 0 if the test statistic lies in the rejection region.Reject H 0 if the test statistic lies in the rejection region.

11. 9-11 Statistical Hypothesis Testing Decision Rule for Two-Tailed Test Decision Rule for Two-Tailed Test Reject H 0 if the test statistic right- tail critical value.Reject H 0 if the test statistic right- tail critical value. Figure 9.2 - Critical value + Critical value

12. 9-12 Statistical Hypothesis Testing Decision Rule for Left-Tailed Test Decision Rule for Left-Tailed Test Reject H 0 if the test statistic < left-tail critical value.Reject H 0 if the test statistic < left-tail critical value. Figure 9.2 - Critical value

13. 9-13 Statistical Hypothesis Testing Decision Rule for Right-Tailed Test Decision Rule for Right-Tailed Test Reject H 0 if the test statistic > right-tail critical value.Reject H 0 if the test statistic > right-tail critical value. + Critical value Figure 9.2

14. 9-14 Statistical Hypothesis Testing Type I Error Type I Error , the probability of a Type I error, is the level of significance (i.e., the probability that the test statistic falls in the rejection region even though H 0 is true). , the probability of a Type I error, is the level of significance (i.e., the probability that the test statistic falls in the rejection region even though H 0 is true).  = P(reject H 0 | H 0 is true) A Type I error is sometimes referred to as a false positive.A Type I error is sometimes referred to as a false positive. For example, if we choose  =.05, we expect to commit a Type I error about 5 times in 100.For example, if we choose  =.05, we expect to commit a Type I error about 5 times in 100.

15. 9-15 Statistical Hypothesis Testing Type I Error Type I Error A small  is desirable, other things being equal.A small  is desirable, other things being equal. Chosen in advance, common choices for  areChosen in advance, common choices for  are.10,.05,.025,.01 and.005 (i.e., 10%, 5%, 2.5%, 1% and.5%). The  risk is the area under the tail(s) of the sampling distribution.The  risk is the area under the tail(s) of the sampling distribution. In a two-sided test, the  risk is split with  /2 in each tail since there are two ways to reject H 0.In a two-sided test, the  risk is split with  /2 in each tail since there are two ways to reject H 0.

16. 9-16 Statistical Hypothesis Testing Type II Error Type II Error , the probability of a type II error, is the probability that the test statistic falls in the acceptance region even though H 0 is false. , the probability of a type II error, is the probability that the test statistic falls in the acceptance region even though H 0 is false.  = P(fail to reject H 0 | H 0 is false)  cannot be chosen in advance because it depends on  and the sample size.  cannot be chosen in advance because it depends on  and the sample size. A small  is desirable, other things being equal.A small  is desirable, other things being equal.

17. 9-17 Statistical Hypothesis Testing Power of a Test Power of a Test The power of a test is the probability that a false hypothesis will be rejected.The power of a test is the probability that a false hypothesis will be rejected. Power = 1 – Power = 1 –  A low  risk means high power.A low  risk means high power. Larger samples lead to increased power.Larger samples lead to increased power. Power = P(reject H 0 | H 0 is false) = 1 – 

18. 9-18 Statistical Hypothesis Testing Relationship Between  and  Relationship Between  and  Both a small  and a small  are desirable.Both a small  and a small  are desirable. For a given type of test and fixed sample size, there is a trade-off between  and .For a given type of test and fixed sample size, there is a trade-off between  and . The larger critical value needed to reduce  risk makes it harder to reject H 0, thereby increasing  risk.The larger critical value needed to reduce  risk makes it harder to reject H 0, thereby increasing  risk. Both  and  can be reduced simultaneously only by increasing the sample size.Both  and  can be reduced simultaneously only by increasing the sample size.

19. 9-19 Statistical Hypothesis Testing Consequences of a Type II Error Consequences of a Type II Error Firms are increasingly wary of Type II errror (failing to recall a product as soon as sample evidence begins to indicate potential problems.)Firms are increasingly wary of Type II errror (failing to recall a product as soon as sample evidence begins to indicate potential problems.) Significance versus Importance Significance versus Importance The standard error of most sample estimators approaches 0 as sample size increases.The standard error of most sample estimators approaches 0 as sample size increases. In this case, no matter how small,  –  0 will be significant if the sample size is large enough.In this case, no matter how small,  –  0 will be significant if the sample size is large enough. Therefore, expect significant effects even when an effect is too slight to have any practical importance.Therefore, expect significant effects even when an effect is too slight to have any practical importance.

20. 9-20 Testing a Mean: Known Population Variance The hypothesized mean  0 that we are testing is a benchmark.The hypothesized mean  0 that we are testing is a benchmark. The value of  0 does not come from a sample.The value of  0 does not come from a sample. The test statistic compares the sample mean x with the hypothesized mean  0.The test statistic compares the sample mean x with the hypothesized mean  0. The difference between x and  0 is divided by the standard error of the mean (denoted  x ).The difference between x and  0 is divided by the standard error of the mean (denoted  x ). The test statistic isThe test statistic is

21. 9-21 Testing a Mean: Known Population Variance Testing the Hypothesis Testing the Hypothesis Step 1: State the hypotheses For example, H 0 :  216 mmStep 1: State the hypotheses For example, H 0 :  216 mm Step 2: Specify the decision rule For example, for  =.05 for the right-tail area, Reject H 0 if z > 1.645, otherwise do notStep 2: Specify the decision rule For example, for  =.05 for the right-tail area, Reject H 0 if z > 1.645, otherwise do not reject H 0

22. 9-22 Testing a Mean: Known Population Variance Testing the Hypothesis Testing the Hypothesis For a two-tailed test, we split the risk of Type I error by putting  /2 in each tail. For example, for  =.05For a two-tailed test, we split the risk of Type I error by putting  /2 in each tail. For example, for  =.05

23. 9-23 Testing a Mean: Known Population Variance Testing the Hypothesis Step 3: Calculate the test statisticStep 3: Calculate the test statistic Step 4: Make the decision If the test statistic falls in the rejection region as defined by the critical value, we reject H 0 and conclude H 1.Step 4: Make the decision If the test statistic falls in the rejection region as defined by the critical value, we reject H 0 and conclude H 1.

24. 9-24 Testing a Mean: Known Population Variance Analogy to Confidence Intervals Analogy to Confidence Intervals A two-tailed hypothesis test at the 5% level of significance (  =.05) is exactly equivalent to asking whether the 95% confidence interval for the mean includes the hypothesized mean.A two-tailed hypothesis test at the 5% level of significance (  =.05) is exactly equivalent to asking whether the 95% confidence interval for the mean includes the hypothesized mean. If the confidence interval includes the hypothesized mean, then we cannot reject the null hypothesis.If the confidence interval includes the hypothesized mean, then we cannot reject the null hypothesis.

25. 9-25 Testing a Mean: Known Population Variance Using the p-Value Approach Using the p-Value Approach The p-value is the probability of the sample result (or one more extreme) assuming that H 0 is true.The p-value is the probability of the sample result (or one more extreme) assuming that H 0 is true. The p-value can be obtained using Excel’s cumulative standard normal function =NORMSDIST(z)The p-value can be obtained using Excel’s cumulative standard normal function =NORMSDIST(z) The p-value can also be obtained from Appendix C-2.The p-value can also be obtained from Appendix C-2. Using the p-value, we reject H 0 if p-value < .Using the p-value, we reject H 0 if p-value < .

26. 9-26 Testing a Mean: Unknown Population Variance When the population standard deviation  is unknown and the population may be assumed normal, the test statistic follows the Student’s t distribution with = n – 1 degrees of freedom.When the population standard deviation  is unknown and the population may be assumed normal, the test statistic follows the Student’s t distribution with = n – 1 degrees of freedom. The test statistic isThe test statistic is Using Student’s t Using Student’s t

27. 9-27 Testing a Mean: Unknown Population Variance Testing a Hypothesis Testing a Hypothesis Step 1: State the hypotheses For example, H 0 :  = 142 H 1 :  ≠ 142Step 1: State the hypotheses For example, H 0 :  = 142 H 1 :  ≠ 142 Step 2: Specify the decision rule For example, for  =.10 for a two-tailed area, Reject H 0 if t > 1.714 or t 1.714 or t < -1.714, otherwise do not reject H 0

28. 9-28 Testing a Mean: Unknown Population Variance Testing a Hypothesis Testing a Hypothesis Step 3: Calculate the test statisticStep 3: Calculate the test statistic Step 4: Make the decision If the test statistic falls in the rejection region as defined by the critical values, we reject H 0 and conclude H 1.Step 4: Make the decision If the test statistic falls in the rejection region as defined by the critical values, we reject H 0 and conclude H 1.

29. 9-29 Testing a Mean: Unknown Population Variance Confidence Intervals versus Hypothesis Test A two-tailed hypothesis test at the 10% level of significance (  =.10) is equivalent to a two-sided 90% confidence interval for the mean.A two-tailed hypothesis test at the 10% level of significance (  =.10) is equivalent to a two-sided 90% confidence interval for the mean. If the confidence interval does not include the hypothesized mean, then we reject the null hypothesis.If the confidence interval does not include the hypothesized mean, then we reject the null hypothesis.

30. 9-30 Testing a Proportion To conduct a hypothesis test, we need to know - the parameter being tested - the sample statistic - the sampling distribution of the sample statisticTo conduct a hypothesis test, we need to know - the parameter being tested - the sample statistic - the sampling distribution of the sample statistic The sampling distribution tells us which test statistic to use.The sampling distribution tells us which test statistic to use. A sample proportion p estimates the population proportion .A sample proportion p estimates the population proportion . Remember that for a large sample, p can be assumed to follow a normal distribution. If so, the test statistic is z.Remember that for a large sample, p can be assumed to follow a normal distribution. If so, the test statistic is z.

31. 9-31 Testing a Proportion p =p =p =p =xn= number of successes sample size z calc = p –  0  p  p If n  0 > 10 and n(1-  0 ) > 10, thenIf n  0 > 10 and n(1-  0 ) > 10, then Where  p =  0 (1-  0 ) n

32. 9-32 Testing a Proportion The value of  0 that we are testing is a benchmark such as past experience, an industry standard, or a product specification.The value of  0 that we are testing is a benchmark such as past experience, an industry standard, or a product specification. The value of  0 does not come from a sample.The value of  0 does not come from a sample.

33. 9-33 Testing a Proportion Critical Value Critical Value The test statistic is compared with a critical value from a table.The test statistic is compared with a critical value from a table. The critical value shows the range of values for the test statistic that would be expected by chance if the H 0 were true.The critical value shows the range of values for the test statistic that would be expected by chance if the H 0 were true. Level of Sig. (  ) Two-tailed test Right-Tailed Test Left-Tailed Test.10 + 1.645 1.282-1.282.05 +1.960 1.645-1.645.01 + 2.576 2.326-2.326

34. 9-34 Testing a Proportion Steps in Testing a Proportion Steps in Testing a Proportion Step 1: State the hypotheses For example, H 0 :  >.13 H 1 : .13 H 1 :  <.13 Step 2: Specify the decision rule For example, for  =.05 for a left-tail area, reject H 0 if z < -1.645, otherwise do notStep 2: Specify the decision rule For example, for  =.05 for a left-tail area, reject H 0 if z < -1.645, otherwise do not reject H 0 Figure 9.12

35. 9-35 Testing a Proportion Steps in Testing a Proportion Steps in Testing a Proportion For a two-tailed test, we split the risk of type I error by putting  /2 in each tail. For example, for  =.05For a two-tailed test, we split the risk of type I error by putting  /2 in each tail. For example, for  =.05 Figure 9.14

36. 9-36 Testing a Proportion Steps in Testing a Proportion Steps in Testing a Proportion Now, check the normality assumption: n  0 > 10 and n(1-  0 ) > 10.Now, check the normality assumption: n  0 > 10 and n(1-  0 ) > 10. Step 3: Calculate the test statisticStep 3: Calculate the test statistic Step 4: Make the decision If the test statistic falls in the rejection region as defined by the critical value, we reject H 0 and conclude H 1.Step 4: Make the decision If the test statistic falls in the rejection region as defined by the critical value, we reject H 0 and conclude H 1. z =z =z =z = p –  0  p  p Where  p =  0 (1-  0 ) n

37. 9-37 Testing a Proportion Using the p-Value Using the p-Value The p-value is the probability of the sample result (or one more extreme) assuming that H 0 is true.The p-value is the probability of the sample result (or one more extreme) assuming that H 0 is true. The p-value can be obtained using Excel’s cumulative standard normal function =NORMSDIST(z)The p-value can be obtained using Excel’s cumulative standard normal function =NORMSDIST(z) The p-value can also be obtained from Appendix C-2.The p-value can also be obtained from Appendix C-2. Using the p-value, we reject H 0 if p-value < .Using the p-value, we reject H 0 if p-value < .

38. 9-38 Testing a Proportion Using the p-Value The p-value is a direct measure of the level of significance at which we could reject H 0.The p-value is a direct measure of the level of significance at which we could reject H 0. Therefore, the smaller the p-value, the more we want to reject H 0.Therefore, the smaller the p-value, the more we want to reject H 0.

39. 9-39 Testing a Proportion Calculating a p-Value for a Two-Tailed Test Calculating a p-Value for a Two-Tailed Test For a two-tailed test, we divide the risk into equal tails. So, to compare the p-value to , first combine the p-values in the two tail areas.For a two-tailed test, we divide the risk into equal tails. So, to compare the p-value to , first combine the p-values in the two tail areas. For example, if our test statistic was -1.975, thenFor example, if our test statistic was -1.975, then 2 x P(z < -1.975) = 2 x.02413 =.04826 At  =.05, we would reject H 0 since p-value =.04826 < .

40. 9-40 Testing a Proportion Effect of  Effect of  No matter which level of significance you use, the test statistic remains the same. For example, for a test statistic of z = 2.152No matter which level of significance you use, the test statistic remains the same. For example, for a test statistic of z = 2.152 - 2.576 2.576 2.576 - 1.645 1.645 1.645 z = 2.152 z = 2.152

41. 9-41 Testing a Proportion Small Samples and Non-Normality Small Samples and Non-Normality In the case where n  0 < 10, use MINITAB to test the hypotheses by finding the exact binomial probability of a sample proportion p. For example,In the case where n  0 < 10, use MINITAB to test the hypotheses by finding the exact binomial probability of a sample proportion p. For example, Figure 9.19

42. 9-42 Power Curves and OC Curves (Optional) Power depends on how far the true value of the parameter is from the null hypothesis value.Power depends on how far the true value of the parameter is from the null hypothesis value. The further away the true population value is from the assumed value, the easier it is for your hypothesis test to detect and the more power it has.The further away the true population value is from the assumed value, the easier it is for your hypothesis test to detect and the more power it has. Remember thatRemember that Power Curves for a Mean Power Curves for a Mean  = P(accept H 0 | H 0 is false) Power = P(reject H 0 | H 0 is false) = 1 – 

43. 9-43 Power Curves and OC Curves We want power to be as close to 1 as possible.We want power to be as close to 1 as possible. The values of  and power will vary, depending on - the difference between the true mean  and the hypothesized mean  0, - the standard deviation, - the sample size n and - the level of significance The values of  and power will vary, depending on - the difference between the true mean  and the hypothesized mean  0, - the standard deviation, - the sample size n and - the level of significance  Power Curves for a Mean Power Curves for a Mean Power = f (  –  0, , n,  )

44. 9-44 Power Curves and OC Curves We can get more power by increasing , but we would then increase the probability of a type I error.We can get more power by increasing , but we would then increase the probability of a type I error. A better way to increase power is to increase the sample size n.A better way to increase power is to increase the sample size n. Power Curves for a Mean Power Curves for a Mean Table 9.8

45. 9-45 Power Curves and OC Curves Calculating Power Calculating Power Step 1: Find the left-tail critical value for the sample mean.Step 1: Find the left-tail critical value for the sample mean. For any given values of , , n, and , and the assumption that X is normally distributed, use the following steps to calculate  and power.For any given values of , , n, and , and the assumption that X is normally distributed, use the following steps to calculate  and power.

46. 9-46 The probability of  error is the area to the right of the critical value x critical which represents P(x > x critical |  =  0 )The probability of  error is the area to the right of the critical value x critical which represents P(x > x critical |  =  0 ) Power Curves and OC Curves Calculating Power Calculating Power The decision rule is: Reject H 0 if x < x criticalThe decision rule is: Reject H 0 if x < x critical

47. 9-47 Power Curves and OC Curves Calculating Power Calculating Power Step 2: Express the difference between the critical value x critical and the true mean  as a z-value:Step 2: Express the difference between the critical value x critical and the true mean  as a z-value: Step 3: Find the  risk and power as areas under the normal curve using Appendix C-2 or Excel.Step 3: Find the  risk and power as areas under the normal curve using Appendix C-2 or Excel.  = P(x > x critical |  =  0 ) Power = P(x < x critical |  =  0 ) = 1 – 

48. 9-48 Power Curves and OC Curves Effect of Sample Size Effect of Sample Size Other things being equal, if sample size were to increase,  risk would decline and power would increase because the critical value x critical would be closer to the hypothesized mean .Other things being equal, if sample size were to increase,  risk would decline and power would increase because the critical value x critical would be closer to the hypothesized mean .

49. 9-49 Power Curves and OC Curves Here is a family of power curves.Here is a family of power curves. Relationship of the Power and OC Curves Relationship of the Power and OC Curves Figure 9.22

50. 9-50 Power Curves and OC Curves The graph of  risk against this same X-axis is called the operating characteristic or OC curve.The graph of  risk against this same X-axis is called the operating characteristic or OC curve. Relationship of the Power and OC Curves Relationship of the Power and OC Curves Figure 9.23

51. 9-51 Power Curves and OC Curves Power Curve for Tests of a Proportion Power Curve for Tests of a Proportion Power depends on - the true proportion (  ), - the hypothesized proportion (  0 ) - the sample size (n) - the level of significance (  )Power depends on - the true proportion (  ), - the hypothesized proportion (  0 ) - the sample size (n) - the level of significance (  ) Table 9.10

52. 9-52 Power Curves and OC Curves Calculating Power Calculating Power Step 1: Find the left-tail critical value for the sample proportion.Step 1: Find the left-tail critical value for the sample proportion. Step 2: Express the difference between the critical value p critical and the true proportion  as a z-value:Step 2: Express the difference between the critical value p critical and the true proportion  as a z-value:

53. 9-53 Power = P(p > p critical |  >  0 ) = 1 –  Power Curves and OC Curves Calculating Power Calculating Power Step 3: Find the  risk and power as areas under the normal curve.Step 3: Find the  risk and power as areas under the normal curve.  = P(p < p critical |  =  0 )

54. 9-54 Power Curves and OC Curves Using LearningStats Using LearningStats Create power curves for a mean or proportion without tedious calculations.Create power curves for a mean or proportion without tedious calculations. Figure 9.25

55. 9-55 Power Curves and OC Curves Using Visual Statistics Using Visual Statistics Create power curves for a mean or proportion without tedious calculations.Create power curves for a mean or proportion without tedious calculations. Figure 9.26

56. 9-56 Reject H 0 if  2  2 upper Tests for One Variance Sometimes we want to compare the variance of a process with a historical benchmark or other standard.Sometimes we want to compare the variance of a process with a historical benchmark or other standard. For a two-tailed test, the hypotheses are: H 0 :  2 =  2 0 H 1 =  2 ≠  2 0For a two-tailed test, the hypotheses are: H 0 :  2 =  2 0 H 1 =  2 ≠  2 0 For a test of one variance, assuming a normal population, the statistic  2 follows the chi- square distribution with degrees of freedom equal to = n – 1.For a test of one variance, assuming a normal population, the statistic  2 follows the chi- square distribution with degrees of freedom equal to = n – 1.  2 = (n – 1)  2    

57. McGraw-Hill/Irwin Copyright © 2009 by The McGraw-Hill Companies, Inc. Applied Statistics in Business & Economics End of Chapter 9 9-57