6 - 1 © 2000 Prentice-Hall, Inc. A First Course in Business Statistics Inferences Based on a Single Sample: Tests of Hypothesis Chapter 6

6 - 2 © 2000 Prentice-Hall, Inc. Learning Objectives 1.Distinguish Types of Hypotheses 2.Describe Hypothesis Testing Process 3. Solve Hypothesis Testing Problems Based on a Single Sample 4.Explain p-Value Concept

6 - 3 © 2000 Prentice-Hall, Inc. What’s a Hypothesis? 1.A Belief about a Population Parameter Parameter Is Population Mean, Proportion, Variance Parameter Is Population Mean, Proportion, Variance Must Be Stated Before Analysis Must Be Stated Before Analysis I believe the mean GPA of this class is 3.5! © T/Maker Co.

6 - 4 © 2000 Prentice-Hall, Inc. Null Hypothesis 1.What Is Tested 2.Has Serious Outcome If Incorrect Decision Made 3.Always Has Equality Sign: , , or  4.Designated H 0 (Pronounced H-oh) 5.Specified as H 0 :   Some Numeric Value Specified with = Sign Even if , or  Specified with = Sign Even if , or  Example, H 0 :   3 Example, H 0 :   3

6 - 5 © 2000 Prentice-Hall, Inc. Alternative Hypothesis 1.Opposite of Null Hypothesis 2.Always Has Inequality Sign: , , or  3.Designated H a 4.Specified H a :  < Some Value Example, H a :  < 3 Example, H a :  < 3

6 - 6 © 2000 Prentice-Hall, Inc. Basic Idea H0H0H0H0 H0H0H0H0 Sampling Distribution

6 - 7 © 2000 Prentice-Hall, Inc. Basic Idea Sampling Distribution It is unlikely that we would get a sample mean of this value H0H0H0H0 H0H0H0H0

6 - 8 © 2000 Prentice-Hall, Inc. Basic Idea Sampling Distribution It is unlikely that we would get a sample mean of this value if in fact this were the population mean 2020 H0H0H0H0 H0H0H0H0

6 - 9 © 2000 Prentice-Hall, Inc. Basic Idea Sampling Distribution It is unlikely that we would get a sample mean of this value if in fact this were the population mean... therefore, we reject the hypothesis that  = H0H0H0H0 H0H0H0H0

© 2000 Prentice-Hall, Inc. Level of Significance 1.Probability 2.Defines Unlikely Values of Sample Statistic if Null Hypothesis Is True Called Rejection Region of Sampling Distribution Called Rejection Region of Sampling Distribution 3.Designated α  (alpha) Typical Values Are.01,.05,.10 Typical Values Are.01,.05,.10 4.Selected by Researcher at Start

© 2000 Prentice-Hall, Inc. Rejection Region ( one-tail test)

© 2000 Prentice-Hall, Inc. Rejection Region (Two-tailed test)

© 2000 Prentice-Hall, Inc. Errors in Making Decision 1.Type I Error Reject True Null Hypothesis Reject True Null Hypothesis Has Serious Consequences Has Serious Consequences Probability of Type I Error Is  (Alpha) Probability of Type I Error Is  (Alpha) Called Level of Significance Called Level of Significance 2.Type II Error Fail to Reject False Null Hypothesis Fail to Reject False Null Hypothesis Probability of Type II Error Is  (Beta) Probability of Type II Error Is  (Beta)

© 2000 Prentice-Hall, Inc. Decision Results H 0 : Innocent

© 2000 Prentice-Hall, Inc.  &  Have an Inverse Relationship   You can’t reduce both errors simultaneously!

© 2000 Prentice-Hall, Inc. H 0 Testing Steps 1.State H 0 and H a 2.Determine Rejection Region (Critical Value) 3.Calculate Test Statistic 4.State Assumptions for Test 5.Conclusion – Reject or Fail to Reject

© 2000 Prentice-Hall, Inc. Overview of Tests Areas of Hypothesis Testing (Like Confidence Intervals) 1.Test about μ σ Known (z), large n 2.Test about μ σ Unknown (t), small n 3.Test about p(large n) ± 3s in interval (0-1)

© 2000 Prentice-Hall, Inc. Two-Tailed Z Test for Mean (  Known) 1.Assumptions Population Is Normally Distributed Population Is Normally Distributed If Not Normal, Can Be Approximated by Normal Distribution (n  30) If Not Normal, Can Be Approximated by Normal Distribution (n  30) 2.Alternative Hypothesis Has  Sign 3.Z-Test Statistic

© 2000 Prentice-Hall, Inc. Two-Tailed Z Test Example Does an average box of cereal contain 368 grams of cereal? A random sample of 25 boxes showed  X = The company has specified  to be 25 grams. Test at the.05 level. 368 gm.

© 2000 Prentice-Hall, Inc. Two-Tailed Z Test Thinking Challenge You’re a Q/C inspector. You want to find out if a new machine is making electrical cords to customer specification: average breaking strength of 70 lb. with  = 3.5 lb. You take a sample of 36 cords & compute a sample mean of 69.7 lb. At the.05 level, is there evidence that the machine is not meeting the average breaking strength?

© 2000 Prentice-Hall, Inc. One-Tailed Z Test for Mean (  Known) 1.Assumptions Population Is Normally Distributed Population Is Normally Distributed If Not Normal, Can Be Approximated by Normal Distribution (n  30) If Not Normal, Can Be Approximated by Normal Distribution (n  30) 2.Alternative Hypothesis Has  or > Sign 3.Z-test Statistic

© 2000 Prentice-Hall, Inc. One-Tailed Z Test Example Does an average box of cereal contain more than 368 grams of cereal? A random sample of 25 boxes showed  X = The company has specified  to be 25 grams. Test at the.05 level. 368 gm.

© 2000 Prentice-Hall, Inc. Observed Significance Levels: p-Values

© 2000 Prentice-Hall, Inc. p-Value 1.Probability of Obtaining a Test Statistic More Extreme (  or  than Actual Sample Value Given H 0 Is True 2.Called Level of Significance Smallest Value of  H 0 Can Be Rejected Smallest Value of  H 0 Can Be Rejected 3.Used to Make ________ Decision If p-Value  , Do Not Reject H 0 If p-Value  , Do Not Reject H 0 If p-Value < , Reject H 0 If p-Value < , Reject H 0

© 2000 Prentice-Hall, Inc. Two-Tailed Z Test p-Value Example Does an average box of cereal contain 368 grams of cereal? A random sample of 25 boxes showed  X = The company has specified  to be 25 grams. Find the p-Value. 368 gm.

© 2000 Prentice-Hall, Inc. One-Tailed Z Test p-Value Example Does an average box of cereal contain more than 368 grams of cereal? A random sample of 25 boxes showed  X = The company has specified  to be 25 grams. Find the p-Value. 368 gm.

© 2000 Prentice-Hall, Inc. t Test for Mean (  Unknown) 1.Assumptions Population Is Normally Distributed Population Is Normally Distributed If Not Normal, Only Slightly Skewed & Large Sample (n  30) Taken If Not Normal, Only Slightly Skewed & Large Sample (n  30) Taken 2.Parametric Test Procedure 3.t Test Statistic

© 2000 Prentice-Hall, Inc. Two-Tailed t Test Thinking Challenge You work for the FTC. A manufacturer of detergent claims that the mean weight of detergent is 3.25 lb. You take a random sample of 64 containers. You calculate the sample average to be lb. with a standard deviation of.117 lb. At the.01 level, is the manufacturer correct? 3.25 lb.

© 2000 Prentice-Hall, Inc. One-Tailed t Test Thinking Challenge You’re a marketing analyst for Wal-Mart. Wal-Mart had teddy bears on sale last week. The weekly sales ($ 00) of bears sold in 10 stores was: At the.05 level, is there evidence that the average bear sales per store is more than 5 ($ 00)?

© 2000 Prentice-Hall, Inc. Z Test of Proportion

© 2000 Prentice-Hall, Inc. Qualitative Data 1.Qualitative Random Variables Yield Responses That Classify e.g., Gender (Male, Female) e.g., Gender (Male, Female) 2.Measurement Reflects # in Category 3.Nominal or Ordinal Scale 4.Examples Do You Own Savings Bonds? Do You Own Savings Bonds? Do You Live On-Campus or Off-Campus? Do You Live On-Campus or Off-Campus?

© 2000 Prentice-Hall, Inc. Proportions 1.Involve Qualitative Variables 2.Fraction or % of Population in a Category 3.If Two Qualitative Outcomes, Binomial Distribution Possess or Don’t Possess Characteristic Possess or Don’t Possess Characteristic 4.Sample Proportion (p) ^

© 2000 Prentice-Hall, Inc. One-Sample Z Test for Proportion 1.Assumptions Two Categorical Outcomes Two Categorical Outcomes Population Follows Binomial Distribution Population Follows Binomial Distribution Normal Approximation Can Be Used Normal Approximation Can Be Used Does Not Contain 0 or n Does Not Contain 0 or n 2.Z-test statistic for proportion Hypothesized population proportion

© 2000 Prentice-Hall, Inc. One-Proportion Z Test Example The present packaging system produces 10% defective cereal boxes. Using a new system, a random sample of 200 boxes had  11 defects. Does the new system produce fewer defects? Test at the.05 level.

© 2000 Prentice-Hall, Inc. One-Proportion Z Test Thinking Challenge You’re an accounting manager. A year-end audit showed 4% of transactions had errors. You implement new procedures. A random sample of 500 transactions had 25 errors. Has the proportion of incorrect transactions changed at the.05 level?