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

2 2 Slide © 2008 Thomson South-Western. All Rights Reserved Chapter 9, Part B Hypothesis Testing Population Proportion Population Proportion Hypothesis Testing and Decision Making Hypothesis Testing and Decision Making Calculating the Probability of Type II Errors Calculating the Probability of Type II Errors Determining the Sample Size for a Determining the Sample Size for a Hypothesis Test About a Population Mean Hypothesis Test About a Population Mean

3 3 Slide © 2008 Thomson South-Western. All Rights Reserved 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

4 4 Slide © 2008 Thomson South-Western. All Rights Reserved n Test Statistic Tests About a Population Proportion where: assuming np > 5 and n (1 – p ) > 5 n Hypothesis tests about a population proportion are based on the difference between the sample proportion p and the hypothesized proportion p 0.

5 5 Slide © 2008 Thomson South-Western. All Rights Reserved 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

6 6 Slide © 2008 Thomson South-Western. All Rights Reserved 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

7 7 Slide © 2008 Thomson South-Western. All Rights Reserved A sample of 120 accidents showed that A sample of 120 accidents showed that 67 were caused by drunk driving. Use these data to test the NSC’s claim with  =.05. Two-Tailed Test About a Population Proportion n Example: National Safety Council (NSC)

8 8 Slide © 2008 Thomson South-Western. All Rights Reserved 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

9 9 Slide © 2008 Thomson South-Western. All Rights Reserved 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

10 Slide © 2008 Thomson South-Western. All Rights Reserved 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 = Determine the criticals value and rejection rule. Reject H 0 if z 1.96 Because > and and < 1.96, we cannot reject H 0.

11 Slide © 2008 Thomson South-Western. All Rights Reserved Hypothesis Testing and Decision Making n When we test a hypothesis, two situations will result: 1. We take no action if the null hypothesis is accepted, but we do take action if the null hypothesis is rejected. Type I error measures the probability of error in incorrectly rejecting the null hypothesis, based on the sample mean, and taking action. A p- value of.02 means we have only a 2% probability of incorrectly rejecting the null hypothesis. 1. We take no action if the null hypothesis is accepted, but we do take action if the null hypothesis is rejected. Type I error measures the probability of error in incorrectly rejecting the null hypothesis, based on the sample mean, and taking action. A p- value of.02 means we have only a 2% probability of incorrectly rejecting the null hypothesis. 2. We take one action if the null hypothesis is accepted, and we take another action if the null hypothesis is rejected. In this situation, Type II error measures the probability of error in incorrectly accepting the null hypothesis, based on the sample mean, and taking action. 2. We take one action if the null hypothesis is accepted, and we take another action if the null hypothesis is rejected. In this situation, Type II error measures the probability of error in incorrectly accepting the null hypothesis, based on the sample mean, and taking action.

12 Slide © 2008 Thomson South-Western. All Rights Reserved Calculating the Probability of a Type II Error in Hypothesis Tests About a Population Mean 1. Formulate the null and alternative hypotheses. 3. Using the rejection rule, solve for the value of the sample mean corresponding to the critical value of sample mean corresponding to the critical value of the test statistic. the test statistic. 2. Using the critical value approach, use the level of significance  to determine the critical value and significance  to determine the critical value and the rejection rule for the test. the rejection rule for the test.

13 Slide © 2008 Thomson South-Western. All Rights Reserved Calculating the Probability of a Type II Error in Hypothesis Tests About a Population Mean 4. Use the results from step 3 to state the values of the sample mean that lead to the acceptance of H 0 ; this defines the acceptance region. sample mean that lead to the acceptance of H 0 ; this defines the acceptance region. 5. Using the sampling distribution of for a value of  satisfying the alternative hypothesis, and the acceptance satisfying the alternative hypothesis, and the acceptance region from step 4, compute the probability that the region from step 4, compute the probability that the sample mean will be in the acceptance region. (This is sample mean will be in the acceptance region. (This is the probability of making a Type II error at the chosen the probability of making a Type II error at the chosen level of .) level of .)

14 Slide © 2008 Thomson South-Western. All Rights Reserved n Example: Metro EMS (revisited) The EMS director wants to The EMS director wants to perform a hypothesis test, with a.05 level of significance, to determine whether or not the service goal of 12 minutes or less is being achieved. Recall that the response times for Recall that the response times for a random sample of 40 medical emergencies were tabulated. The sample mean is minutes. The population standard deviation is believed to be 3.2 minutes. Calculating the Probability of a Type II Error

15 Slide © 2008 Thomson South-Western. All Rights Reserved 4. We will accept H 0 when x < Value of the sample mean that identifies the rejection region: the rejection region: 2. Rejection rule is: Reject H 0 if z > Hypotheses are: H 0 :  and H a :  Calculating the Probability of a Type II Error

16 Slide © 2008 Thomson South-Western. All Rights Reserved Values of  1-  5. Probabilities that the sample mean will be in the acceptance region: in the acceptance region: Calculating the Probability of a Type II Error

17 Slide © 2008 Thomson South-Western. All Rights Reserved Calculating the Probability of a Type II Error n Calculating the Probability of a Type II Error When the true population mean  is far above When the true population mean  is far above the null hypothesis value of 12, there is a low the null hypothesis value of 12, there is a low probability that we will make a Type II error. probability that we will make a Type II error. When the true population mean  is close to When the true population mean  is close to the null hypothesis value of 12, there is a high the null hypothesis value of 12, there is a high probability that we will make a Type II error. probability that we will make a Type II error. Observations about the preceding table: Example:  = ,  =.9500 Example:  = 14.0,  =.0104

18 Slide © 2008 Thomson South-Western. All Rights Reserved Power of the Test n The probability of correctly rejecting H 0 when it is false is called the power of the test. 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 – . n We can show graphically the power associated with each value of  ; such a graph is called a with each value of  ; such a graph is called a power curve. (See next slide.) power curve. (See next slide.) What value should we use for  If performance is acceptable when the actual population mean is within the test limit(s), although the population mean does not equal the hypothesized mean, then use the closest test limit to calculate the power of the test. What value should we use for  If performance is acceptable when the actual population mean is within the test limit(s), although the population mean does not equal the hypothesized mean, then use the closest test limit to calculate the power of the test.

19 Slide © 2008 Thomson South-Western. All Rights Reserved Power Curve  H 0 False

20 Slide © 2008 Thomson South-Western. All Rights Reserved n The specified level of significance determines the probability of making a Type I error. probability of making a Type I error. By controlling the sample size, the probability of By controlling the sample size, the probability of making a Type II error is controlled. making a Type II error is controlled. Determining the Sample Size for a Hypothesis Test About a Population Mean

21 Slide © 2008 Thomson South-Western. All Rights Reserved 00 00   aa aa Sampling distribution of when H 0 is true and  =  0 Sampling distribution of when H 0 is true and  =  0 Sampling distribution of when H 0 is false and  a >  0 Sampling distribution of when H 0 is false and  a >  0 Reject H 0  cc H 0 :   H a :   Note: Determining the Sample Size for a Hypothesis Test About a Population Mean

22 Slide © 2008 Thomson South-Western. All Rights Reserved 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 Determining the Sample Size for a One-Tail Hypothesis Test About a Population Mean Note: In a two-tailed hypothesis test, use z  /2 not z 

23 Slide © 2008 Thomson South-Western. All Rights Reserved 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. If you are concerned only about Type I error and not Type II error, then use small values for . If you are concerned about Type II error but you are not testing for Type II error, then do not use very small values for  If you are concerned only about Type I error and not Type II error, then use small values for . If you are concerned about Type II error but you are not testing for Type II error, then do not use very small values for 

24 Slide © 2008 Thomson South-Western. All Rights Reserved Determining the Sample Size for a Hypothesis Test About a Population Mean n Let’s assume that the director of medical services makes the following statements about the services makes the following statements about the allowable probabilities for the Type I and Type II allowable probabilities for the Type I and Type II errors: errors: If the mean response time is  = 12 minutes, I am willing to risk an  =.05 probability of rejecting H 0.If the mean response time is  = 12 minutes, I am willing to risk an  =.05 probability of rejecting H 0. If the mean response time is 0.75 minutes over the specification (  = 12.75), I am willing to risk a  =.10 probability of not rejecting H 0.If the mean response time is 0.75 minutes over the specification (  = 12.75), I am willing to risk a  =.10 probability of not rejecting H 0.

25 Slide © 2008 Thomson South-Western. All Rights Reserved Determining the Sample Size for a Hypothesis Test About a Population Mean  =.05,  =.10 z  = 1.645, z  = 1.28  0 = 12,  a =  = 3.2

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