Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-1 Business Statistics, 4e by Ken Black Chapter 9 Statistical Inference: Hypothesis Testing for Single Populations

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-2 Learning Objectives Understand the logic of hypothesis testing, and know how to establish null and alternate hypotheses. Understand Type I and Type II errors, and know how to solve for Type II errors. Know how to implement the HTAB system to test hypotheses. Test hypotheses about a single population mean when  is known. Test hypotheses about a single population mean when  is unknown. Test hypotheses about a single population proportion. Test hypotheses about a single population variance.

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-3 Types of Hypotheses Research Hypothesis –a statement of what the researcher believes will be the outcome of an experiment or a study. Statistical Hypotheses –a more formal structure derived from the research hypothesis. Substantive Hypotheses –a statistically significant difference does not imply or mean a material, substantive difference.

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-4 Example Research Hypotheses Older workers are more loyal to a company Companies with more than $1 billion of assets spend a higher percentage of their annual budget on advertising than do companies with less than $1 billion of assets. The price of scrap metal is a good indicator of the industrial production index six months later.

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-5 Statistical Hypotheses Two Parts –a null hypothesis –an alternative hypothesis Null Hypothesis – nothing new is happening Alternative Hypothesis – something new is happening Notation –null: H 0 –alternative: H a

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-6 Null and Alternative Hypotheses The Null and Alternative Hypotheses are mutually exclusive. Only one of them can be true. The Null and Alternative Hypotheses are collectively exhaustive. They are stated to include all possibilities. (An abbreviated form of the null hypothesis is often used.) The Null Hypothesis is assumed to be true. The burden of proof falls on the Alternative Hypothesis.

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-7 Null and Alternative Hypotheses: Example A manufacturer is filling 40 oz. packages with flour. The company wants the package contents to average 40 ounces.

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-8 One-tailed Tests One-tailed and Two-tailed Tests Two-tailed Test

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-9 HTAB System to Test Hypotheses Task 1: HYPOTHESIZE Task 2: TEST Task 3: TAKE STATISTICAL ACTION Task 4: DETERMINING THE BUSINESS IMPLICATIONS

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons Steps in Testing Hypotheses 1. Establish hypotheses: state the null and alternative hypotheses. 2. Determine the appropriate statistical test and sampling distribution. 3. Specify the Type I error rate (  4. State the decision rule. 5. Gather sample data. 6. Calculate the value of the test statistic. 7. State the statistical conclusion. 8. Make a managerial decision.

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons HTAB Paradigm – Task 1 Task 1: Hypotheses Step 1. Establish hypotheses: state the null and alternative hypotheses.

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons HTAB Paradigm – Task 2 Task 2: Test Step 2. Determine the appropriate statistical test and sampling distribution. Step 3. Specify the Type I error rate (  Step 4. State the decision rule. Step 5. Gather sample data. Step 6. Calculate the value of the test statistic.

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons HTAB Paradigm – Task 3 Task 3: Take Statistical Action Step 7. State the statistical conclusion.

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons HTAB Paradigm – Task 4 Task 4: Determine the business implications Step 8. Make a managerial decision.

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons Rejection and Non Rejection Regions  =40 oz Non Rejection Region Rejection Region Critical Value Rejection Region Critical Value

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons Type I and Type II Errors Type I Error –Rejecting a true null hypothesis –The probability of committing a Type I error is called , the level of significance. Type II Error –Failing to reject a false null hypothesis –The probability of committing a Type II error is called .

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons Decision Table for Hypothesis Testing ( () Null TrueNull False Fail to reject null Correct Decision Type II error  ) Reject nullType I error  Correct Decision

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons One-tailed Tests  =40 oz Rejection Region Non Rejection Region Critical Value  =40 oz Rejection Region Non Rejection Region Critical Value

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons Two-tailed Tests  =12 oz Rejection Region Non Rejection Region Critical Values Rejection Region

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons CPA Net Income Example: Two-tailed Test (Part 1) Rejection Region Non Rejection Region  =0 Rejection Region

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons CPA Net Income Example: Two-tailed Test (Part 2)

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons CPA Net Income Example: Critical Value Method (Part 1) Rejection Region Non Rejection Region  =0 Rejection Region 72,22377,605

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons CPA Net Income Example: Critical Value Method (Part 2) Rejection Region Non Rejection Region  =0 Rejection Region 72,22377,605

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons Demonstration Problem 9.1: z Test (Part 1) Rejection Region Non Rejection Region 0  =.05

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons Demonstration Problem 9.1: z Test (Part 2) Rejection Region Non Rejection Region 0  =.05

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons Demonstration Problem 9.1: Critical Value (Part 1) Rejection Region Non Rejection Region 0  =

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons Demonstration Problem 9.1: Critical Value (Part 2) Rejection Region Non Rejection Region 0  =

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons Rejection Region Non Rejection Region 0  =.05 Demonstration Problem 9.1: Using the p-Value

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons Demonstration Problem 9.1: MINITAB Test of mu = vs mu < The assumed sigma = VariableNMEANSTDEVSE MEANZP VALUE Ratings

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons Demonstration Problem 9.1: Excel (Part 1)

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons Demonstration Problem 9.1: Excel (Part 2) H 0 :  = 4.3 H a :  < n ==COUNT(A4:H7)  = 0.05 Mean ==AVERAGE(A4:H7) S ==STDEV(A4:H7) Std Error ==B12/SQRT(B9) Z ==(B11-B1)/B13 p-Value=NORMSDIST(B14)

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons Two-tailed Test:  Unknown,  =.05 (Part 1) Weights in Pounds of a Sample of 20 Plates

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons Two-tailed Test:  Unknown,  =.05 (part 2) Critical Values Non Rejection Region Rejection Regions

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons Two-tailed Test:  Unknown,  =.05 (part 3) Critical Values Non Rejection Region Rejection Regions

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons MINITAB Computer Printout for the Machine Plate Example Test of mu = vs mu not = VariableNMEANSTDEVSE MEANTP VALUE Platewt

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons Machine Plate Example: Excel (Part 1)

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons Machine Plate Example: Excel (Part 2) AB CDE 1 H 0 :  = 25 2 H a :   n ==COUNT(A4:E7) 10  = Mean ==AVERAGE(A4:E7) 12 S ==STDEV(A4:E7) 13 Std Error ==B12/SQRT(B9) 14 t ==(B11-B1)/B13 15 p-Value=TDIST(B14,B9-1,2)

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons Demonstration Problem 9.2 (Part 1) Size in Acres of 23 Farms

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons Demonstration Problem 9.2 (Part 2) Critical Value Non Rejection Region Rejection Region

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons Demonstration Problem 9.2 (Part 3) Critical Value Non Rejection Region Rejection Region

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons z Test of Population Proportion

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons Testing Hypotheses about a Proportion: Manufacturer Example (Part 1) Critical Values Non Rejection Region Rejection Regions

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons Testing Hypotheses about a Proportion: Manufacturer Example (Part 2) Critical Values Non Rejection Region Rejection Regions

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons Demonstration Problem 9.3 (Part 1) Critical Value Non Rejection Region Rejection Region

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons Demonstration Problem 9.3 (Part 2) Critical Value Non Rejection Region Rejection Region

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons Hypothesis Test for  2: Demonstration Problem 9.4 (Part 1) 0 df =

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons Hypothesis Test for  2: Demonstration Problem 9.4 (Part 2) 0 df =

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons Solving for Type II Errors: The Beverage Example Rejectio n Region Non Rejection Region  =0  =.05

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons Type II Error for Beverage Example with  =11.99 oz  =.05 Reject H o Do Not Reject H o   H o is True H o is False 95%  =.8023 Correct Decision Type I Error Type II Error Correct Decision 19.77%    Z0Z0 Z1Z1

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons Type II Error for Demonstration Problem 9.5, with  =11.96 oz  =.05  H o is True H o is False 95%  Reject H o Do Not Reject H o  =.0708 Correct Decision Type I Error Type II Error Correct Decision 92.92%   Z0Z0 Z1Z1

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons  Values and Power Values for the Soft-Drink Example  Power

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons Operating Characteristic Curve for the Soft-Drink Example Probability 

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons Power Curve for the Soft-Drink Example Probability 