8 - 1 © 1998 Prentice-Hall, Inc. Chapter 8 Inferences Based on a Single Sample: Tests of Hypothesis

8 - 2 © 1998 Prentice-Hall, Inc. Learning Objectives 1.Distinguish types of hypotheses 2.Describe hypothesis testing process 3.Explain p-value concept 4.Solve hypothesis testing problems based on a single sample

8 - 3 © 1998 Prentice-Hall, Inc. Types of Statistical Applications

8 - 4 © 1998 Prentice-Hall, Inc. Hypothesis Testing Concepts

8 - 5 © 1998 Prentice-Hall, Inc. Hypothesis Testing

8 - 6 © 1998 Prentice-Hall, Inc. Hypothesis Testing Population

8 - 7 © 1998 Prentice-Hall, Inc. Hypothesis Testing Population I believe the population mean age is 50 (hypothesis).

8 - 8 © 1998 Prentice-Hall, Inc. Hypothesis Testing Population I believe the population mean age is 50 (hypothesis). Mean  X = 20 Random sample

8 - 9 © 1998 Prentice-Hall, Inc. Hypothesis Testing Population I believe the population mean age is 50 (hypothesis). Mean  X = 20 Reject hypothesis! Not close. Random sample

© 1998 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

© 1998 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.

© 1998 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 5.Specified as H 0 :   Some numeric value Written with = sign even if , or  Written with = sign even if , or  Example, H 0 :   3 Example, H 0 :   3

© 1998 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

© 1998 Prentice-Hall, Inc. Identifying Hypotheses Steps

© 1998 Prentice-Hall, Inc. Identifying Hypotheses Steps Steps 1.State question statistically

© 1998 Prentice-Hall, Inc. Identifying Hypotheses Steps Steps 1.State question statistically Example Is the population mean different from 3?

© 1998 Prentice-Hall, Inc. Identifying Hypotheses Steps Steps 1.State question statistically Example Is the population mean different from 3? 1.   3

© 1998 Prentice-Hall, Inc. Identifying Hypotheses Steps Steps 1.State question statistically 2.State opposite statistically Must be mutually exclusive & exhaustive Must be mutually exclusive & exhaustiveExample Is the population mean different from 3? 1.   3

© 1998 Prentice-Hall, Inc. Identifying Hypotheses Steps Steps 1.State question statistically 2.State opposite statistically Must be mutually exclusive & exhaustive Must be mutually exclusive & exhaustiveExample Is the population mean different from 3? 1.   3 2.  = 3

© 1998 Prentice-Hall, Inc. Identifying Hypotheses Steps Steps 1.State question statistically 2.State opposite statistically Must be mutually exclusive & exhaustive Must be mutually exclusive & exhaustive 3.Select & state alternative hypothesis Has the , sign Has the , signExample Is the population mean different from 3? 1.   3 2.  = 3

© 1998 Prentice-Hall, Inc. Identifying Hypotheses Steps Steps 1.State question statistically 2.State opposite statistically Must be mutually exclusive & exhaustive Must be mutually exclusive & exhaustive 3.Select & state alternative hypothesis Has the , sign Has the , signExample Is the population mean different from 3? 1.   3 2.  = 3 3. H a :   3

© 1998 Prentice-Hall, Inc. Identifying Hypotheses Steps Steps 1.State question statistically 2.State opposite statistically Must be mutually exclusive & exhaustive Must be mutually exclusive & exhaustive 3.Select & state alternative hypothesis Has the , sign Has the , sign 4.State null hypothesis Example Is the population mean different from 3? 1.   3 2.  = 3 3. H a :   3

© 1998 Prentice-Hall, Inc. Identifying Hypotheses Steps Steps 1.State question statistically 2.State opposite statistically Must be mutually exclusive & exhaustive Must be mutually exclusive & exhaustive 3.Select & state alternative hypothesis Has the , sign Has the , sign 4.State null hypothesis Example Is the population mean different from 3? 1.   3 2.  = 3 3. H a :   3 4. H 0 :  = 3

© 1998 Prentice-Hall, Inc. Identifying Hypotheses Example 1 Steps 1.State question statistically 2.State opposite statistically Must be mutually exclusive & exhaustive Must be mutually exclusive & exhaustive 3.Select & state alternative hypothesis Has the , sign Has the , sign 4.State null hypothesis Example Is the population average amount of TV viewing 12 hours?

© 1998 Prentice-Hall, Inc. Identifying Hypotheses Example 1 Steps 1.State question statistically 2.State opposite statistically Must be mutually exclusive & exhaustive Must be mutually exclusive & exhaustive 3.Select & state alternative hypothesis Has the , sign Has the , sign 4.State null hypothesis Example Is the population average amount of TV viewing 12 hours? 1.  =

© 1998 Prentice-Hall, Inc. Identifying Hypotheses Example 1 Steps 1.State question statistically 2.State opposite statistically Must be mutually exclusive & exhaustive Must be mutually exclusive & exhaustive 3.Select & state alternative hypothesis Has the , sign Has the , sign 4.State null hypothesis Example Is the population average amount of TV viewing 12 hours? 1.  =  

© 1998 Prentice-Hall, Inc. Identifying Hypotheses Example 1 Steps 1.State question statistically 2.State opposite statistically Must be mutually exclusive & exhaustive Must be mutually exclusive & exhaustive 3.Select & state alternative hypothesis Has the , sign Has the , sign 4.State null hypothesis Example Is the population average amount of TV viewing 12 hours? 1.  =   H a :   12 4.

© 1998 Prentice-Hall, Inc. Identifying Hypotheses Example 1 Steps 1.State question statistically 2.State opposite statistically Must be mutually exclusive & exhaustive Must be mutually exclusive & exhaustive 3.Select & state alternative hypothesis Has the , sign Has the , sign 4.State null hypothesis Example Is the population average amount of TV viewing 12 hours? 1.  =   H a :   H 0 :  = 12

© 1998 Prentice-Hall, Inc. Identifying Hypotheses Example 2 Steps 1.State question statistically 2.State opposite statistically Must be mutually exclusive & exhaustive Must be mutually exclusive & exhaustive 3.Select & state alternative hypothesis Has the , sign Has the , sign 4.State null hypothesis Example Is the population average amount of TV viewing different from 12 hours?

© 1998 Prentice-Hall, Inc. Identifying Hypotheses Example 2 Steps 1.State question statistically 2.State opposite statistically Must be mutually exclusive & exhaustive Must be mutually exclusive & exhaustive 3.Select & state alternative hypothesis Has the , sign Has the , sign 4.State null hypothesis Example Is the population average amount of TV viewing different from 12 hours? 1.  

© 1998 Prentice-Hall, Inc. Identifying Hypotheses Example 2 Steps 1.State question statistically 2.State opposite statistically Must be mutually exclusive & exhaustive Must be mutually exclusive & exhaustive 3.Select & state alternative hypothesis Has the , sign Has the , sign 4.State null hypothesis Example Is the population average amount of TV viewing different from 12 hours? 1.    =

© 1998 Prentice-Hall, Inc. Identifying Hypotheses Example 2 Steps 1.State question statistically 2.State opposite statistically Must be mutually exclusive & exhaustive Must be mutually exclusive & exhaustive 3.Select & state alternative hypothesis Has the , sign Has the , sign 4.State null hypothesis Example Is the population average amount of TV viewing different from 12 hours? 1.    = H a :   12 4.

© 1998 Prentice-Hall, Inc. Identifying Hypotheses Example 2 Steps 1.State question statistically 2.State opposite statistically Must be mutually exclusive & exhaustive Must be mutually exclusive & exhaustive 3.Select & state alternative hypothesis Has the , sign Has the , sign 4.State null hypothesis Example Is the population average amount of TV viewing different from 12 hours? 1.    = H a :   H 0 :  = 12

© 1998 Prentice-Hall, Inc. Identifying Hypotheses Example 3 Steps 1.State question statistically 2.State opposite statistically Must be mutually exclusive & exhaustive Must be mutually exclusive & exhaustive 3.Select & state alternative hypothesis Has the , sign Has the , sign 4.State null hypothesis Example Is the average cost per hat less than or equal to $20?

© 1998 Prentice-Hall, Inc. Identifying Hypotheses Example 3 Steps 1.State question statistically 2.State opposite statistically Must be mutually exclusive & exhaustive Must be mutually exclusive & exhaustive 3.Select & state alternative hypothesis Has the , sign Has the , sign 4.State null hypothesis Example Is the average cost per hat less than or equal to $20? 1.  

© 1998 Prentice-Hall, Inc. Identifying Hypotheses Example 3 Steps 1.State question statistically 2.State opposite statistically Must be mutually exclusive & exhaustive Must be mutually exclusive & exhaustive 3.Select & state alternative hypothesis Has the , sign Has the , sign 4.State null hypothesis Example Is the average cost per hat less than or equal to $20? 1.   

© 1998 Prentice-Hall, Inc. Identifying Hypotheses Example 3 Steps 1.State question statistically 2.State opposite statistically Must be mutually exclusive & exhaustive Must be mutually exclusive & exhaustive 3.Select & state alternative hypothesis Has the , sign Has the , sign 4.State null hypothesis Example Is the average cost per hat less than or equal to $20? 1.    H a :   20 4.

© 1998 Prentice-Hall, Inc. Identifying Hypotheses Example 3 Steps 1.State question statistically 2.State opposite statistically Must be mutually exclusive & exhaustive Must be mutually exclusive & exhaustive 3.Select & state alternative hypothesis Has the , sign Has the , sign 4.State null hypothesis Example Is the average cost per hat less than or equal to $20? 1.    H a :   H 0 :  = 20

© 1998 Prentice-Hall, Inc. Identifying Hypotheses Example 4 Steps 1.State question statistically 2.State opposite statistically Must be mutually exclusive & exhaustive Must be mutually exclusive & exhaustive 3.Select & state alternative hypothesis Has the , sign Has the , sign 4.State null hypothesis Example Is the average amount spent in the bookstore greater than $25?

© 1998 Prentice-Hall, Inc. Identifying Hypotheses Example 4 Steps 1.State question statistically 2.State opposite statistically Must be mutually exclusive & exhaustive Must be mutually exclusive & exhaustive 3.Select & state alternative hypothesis Has the , sign Has the , sign 4.State null hypothesis Example Is the average amount spent in the bookstore greater than $25? 1.  

© 1998 Prentice-Hall, Inc. Identifying Hypotheses Example 4 Steps 1.State question statistically 2.State opposite statistically Must be mutually exclusive & exhaustive Must be mutually exclusive & exhaustive 3.Select & state alternative hypothesis Has the , sign Has the , sign 4.State null hypothesis Example Is the average amount spent in the bookstore greater than $25? 1.   

© 1998 Prentice-Hall, Inc. Identifying Hypotheses Example 4 Steps 1.State question statistically 2.State opposite statistically Must be mutually exclusive & exhaustive Must be mutually exclusive & exhaustive 3.Select & state alternative hypothesis Has the , sign Has the , sign 4.State null hypothesis Example Is the average amount spent in the bookstore greater than $25? 1.    H a :   25 4.

© 1998 Prentice-Hall, Inc. Identifying Hypotheses Example 4 Steps 1.State question statistically 2.State opposite statistically Must be mutually exclusive & exhaustive Must be mutually exclusive & exhaustive 3.Select & state alternative hypothesis Has the , sign Has the , sign 4.State null hypothesis Example Is the average amount spent in the bookstore greater than $25? 1.    H a :   H 0 :  = 25

© 1998 Prentice-Hall, Inc. Basic Idea

© 1998 Prentice-Hall, Inc. Basic Idea H0H0H0H0 H0H0H0H0 Sampling Distribution

© 1998 Prentice-Hall, Inc. Basic Idea Sampling Distribution It is unlikely that we would get a sample mean of this value H0H0H0H0 H0H0H0H0

© 1998 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

© 1998 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

© 1998 Prentice-Hall, Inc. Level of Significance 1.Defines unlikely values of sample statistic if null hypothesis is true Called rejection region of sampling distribution Called rejection region of sampling distribution 2.Is a probability 3.Denoted  (alpha) 4.Selected by researcher at start Typical values are.01,.05,.10 Typical values are.01,.05,.10

© 1998 Prentice-Hall, Inc. Rejection Region (One-Tail Test)

© 1998 Prentice-Hall, Inc. Ho Value Sample Statistic Rejection Region (One-Tail Test) Sampling Distribution

© 1998 Prentice-Hall, Inc. Ho Value Sample Statistic Rejection Region Rejection Region (One-Tail Test) Sampling Distribution

© 1998 Prentice-Hall, Inc. Ho Value Sample Statistic Rejection Region Nonrejection Region Rejection Region (One-Tail Test) Sampling Distribution

© 1998 Prentice-Hall, Inc. Ho Value Sample Statistic Rejection Region Nonrejection Region Rejection Region (One-Tail Test) Sampling Distribution Critical Value

© 1998 Prentice-Hall, Inc. Ho Value   Sample Statistic Rejection Region Nonrejection Region Rejection Region (One-Tail Test) Sampling Distribution Critical Value

© 1998 Prentice-Hall, Inc. Rejection Region (One-Tail Test) Sampling Distribution 1 -  Level of Confidence

© 1998 Prentice-Hall, Inc. Rejection Region (One-Tail Test) Sampling Distribution 1 -  Level of Confidence Observed sample statistic

© 1998 Prentice-Hall, Inc. Rejection Region (One-Tail Test) Sampling Distribution 1 -  Level of Confidence

© 1998 Prentice-Hall, Inc. Rejection Regions (Two-Tailed Test)

© 1998 Prentice-Hall, Inc. Ho Value Sample Statistic Rejection Regions (Two-Tailed Test) Sampling Distribution

© 1998 Prentice-Hall, Inc. Ho Value Sample Statistic Rejection Region Rejection Region Rejection Regions (Two-Tailed Test) Sampling Distribution

© 1998 Prentice-Hall, Inc. Ho Value Sample Statistic Rejection Region Rejection Region Nonrejection Region Rejection Regions (Two-Tailed Test) Sampling Distribution

© 1998 Prentice-Hall, Inc. Ho Value Critical Value Critical Value Sample Statistic Rejection Region Rejection Region Nonrejection Region Rejection Regions (Two-Tailed Test) Sampling Distribution

© 1998 Prentice-Hall, Inc. Ho Value Critical Value Critical Value 1/2     Sample Statistic Rejection Region Rejection Region Nonrejection Region Rejection Regions (Two-Tailed Test) Sampling Distribution

© 1998 Prentice-Hall, Inc. Rejection Regions (Two-Tailed Test) Sampling Distribution 1 -  Level of Confidence

© 1998 Prentice-Hall, Inc. Rejection Regions (Two-Tailed Test) Sampling Distribution 1 -  Level of Confidence

© 1998 Prentice-Hall, Inc. Rejection Regions (Two-Tailed Test) Sampling Distribution 1 -  Level of Confidence

© 1998 Prentice-Hall, Inc. Rejection Regions (Two-Tailed Test) Sampling Distribution 1 -  Level of Confidence

© 1998 Prentice-Hall, Inc. Decision Making Risks

© 1998 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 Do not reject false null hypothesis Do not reject false null hypothesis Probability of Type II error is  (beta) Probability of Type II error is  (beta)

© 1998 Prentice-Hall, Inc. Decision Results H 0 : Innocent

© 1998 Prentice-Hall, Inc. Decision Results H 0 : Innocent

© 1998 Prentice-Hall, Inc. Hypothesis Testing Steps

© 1998 Prentice-Hall, Inc. H 0 Testing Steps

© 1998 Prentice-Hall, Inc. H 0 Testing Steps n State H 0 n State H 1 Choose  Choose  n Choose n n Choose test

© 1998 Prentice-Hall, Inc. H 0 Testing Steps n Set up critical values n Collect data n Compute test statistic n Make statistical decision n Express decision n State H 0 n State H 1 Choose  Choose  n Choose n n Choose test

© 1998 Prentice-Hall, Inc. One Population Tests

© 1998 Prentice-Hall, Inc. One Population Tests One Population

© 1998 Prentice-Hall, Inc. One Population Tests One Population Mean

© 1998 Prentice-Hall, Inc. One Population Tests One Population MeanProportion

© 1998 Prentice-Hall, Inc. One Population Tests One Population Z Test (1 & 2 tail) MeanProportion LargeLarge SampleSample

© 1998 Prentice-Hall, Inc. One Population Tests One Population Z Test (1 & 2 tail) t Test (1 & 2 tail) MeanProportion LargeLarge SampleSample SmallSmall SampleSample

© 1998 Prentice-Hall, Inc. One Population Tests

© 1998 Prentice-Hall, Inc. Two-Tailed Z Test of Mean (Large Sample)

© 1998 Prentice-Hall, Inc. One Population Tests

© 1998 Prentice-Hall, Inc. Two-Tailed Z Test for Mean (Large Sample) 1.Assumptions Sample size at least 30 (n  30) Sample size at least 30 (n  30) If population standard deviation unknown, use sample standard deviation If population standard deviation unknown, use sample standard deviation 2.Alternative hypothesis has  sign

© 1998 Prentice-Hall, Inc. Two-Tailed Z Test for Mean (Large Sample) 1.Assumptions Sample size at least 30 (n  30) Sample size at least 30 (n  30) If population standard deviation unknown, use sample standard deviation If population standard deviation unknown, use sample standard deviation 2.Alternative hypothesis has  sign 3.Z-test statistic

© 1998 Prentice-Hall, Inc. Two-Tailed Z Test Example Does an average box of cereal contain 368 grams of cereal? A random sample of 36 boxes showed  X = The company has specified  to be 25 grams. Test at the.05 level. 368 gm.

© 1998 Prentice-Hall, Inc. Two-Tailed Z Test Solution H 0 : H a :   n  Critical Value(s): Test Statistic: Decision:Conclusion:

© 1998 Prentice-Hall, Inc. Two-Tailed Z Test Solution H 0 :  = 368 H a :   368   n  Critical Value(s): Test Statistic: Decision:Conclusion:

© 1998 Prentice-Hall, Inc. Two-Tailed Z Test Solution H 0 :  = 368 H a :   368  .05 n  36 Critical Value(s): Test Statistic: Decision:Conclusion:

© 1998 Prentice-Hall, Inc. Two-Tailed Z Test Solution H 0 :  = 368 H a :   368  .05 n  36 Critical Value(s): Test Statistic: Decision:Conclusion:

© 1998 Prentice-Hall, Inc. Two-Tailed Z Test Solution H 0 :  = 368 H a :   368  .05 n  36 Critical Value(s): Test Statistic: Decision:Conclusion:

© 1998 Prentice-Hall, Inc. Two-Tailed Z Test Solution H 0 :  = 368 H a :   368  .05 n  36 Critical Value(s): Test Statistic: Decision:Conclusion: Do not reject at  =.05

© 1998 Prentice-Hall, Inc. Two-Tailed Z Test Solution H 0 :  = 368 H a :   368  .05 n  36 Critical Value(s): Test Statistic: Decision:Conclusion: Do not reject at  =.05 No evidence average is not 368

© 1998 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? AloneGroupClass

© 1998 Prentice-Hall, Inc. Two-Tailed Z Test Solution* H 0 : H a :  = n = Critical Value(s): Test Statistic: Decision:Conclusion:

© 1998 Prentice-Hall, Inc. Two-Tailed Z Test Solution* H 0 :  = 70 H a :   70  = n = Critical Value(s): Test Statistic: Decision:Conclusion:

© 1998 Prentice-Hall, Inc. Two-Tailed Z Test Solution* H 0 :  = 70 H a :   70  =.05 n = 36 Critical Value(s): Test Statistic: Decision:Conclusion:

© 1998 Prentice-Hall, Inc. Two-Tailed Z Test Solution* H 0 :  = 70 H a :   70  =.05 n = 36 Critical Value(s): Test Statistic: Decision:Conclusion:

© 1998 Prentice-Hall, Inc. Two-Tailed Z Test Solution* H 0 :  = 70 H a :   70  =.05 n = 36 Critical Value(s): Test Statistic: Decision:Conclusion:

© 1998 Prentice-Hall, Inc. Two-Tailed Z Test Solution* H 0 :  = 70 H a :   70  =.05 n = 36 Critical Value(s): Test Statistic: Decision:Conclusion: Do not reject at  =.05

© 1998 Prentice-Hall, Inc. Two-Tailed Z Test Solution* H 0 :  = 70 H a :   70  =.05 n = 36 Critical Value(s): Test Statistic: Decision:Conclusion: Do not reject at  =.05 No evidence average is not 70

© 1998 Prentice-Hall, Inc. One-Tailed Z Test of Mean (Large Sample)

© 1998 Prentice-Hall, Inc. One-Tailed Z Test for Mean (Large Sample) 1.Assumptions Sample size at least 30 (n  30) Sample size at least 30 (n  30) If population standard deviation unknown, use sample standard deviation If population standard deviation unknown, use sample standard deviation 2.Alternative hypothesis has sign

© 1998 Prentice-Hall, Inc. One-Tailed Z Test for Mean (Large Sample) 1.Assumptions Sample size at least 30 (n  30) Sample size at least 30 (n  30) If population standard deviation unknown, use sample standard deviation If population standard deviation unknown, use sample standard deviation 2.Alternative hypothesis has  or > sign 3.Z-test statistic

© 1998 Prentice-Hall, Inc. One-Tailed Z Test for Mean Hypotheses

© 1998 Prentice-Hall, Inc. One-Tailed Z Test for Mean Hypotheses H 0 :  =  0 H a :  < 0 Must be significantly below 

© 1998 Prentice-Hall, Inc. One-Tailed Z Test for Mean Hypotheses H 0 :  =  0 H a :  < 0 H 0 :  =  0 H a :  > 0 Must be significantly below  Small values satisfy H 0. Don’t reject!

© 1998 Prentice-Hall, Inc. One-Tailed Z Test Finding Critical Z

© 1998 Prentice-Hall, Inc. One-Tailed Z Test Finding Critical Z What is Z given  =.025?  =.025 

© 1998 Prentice-Hall, Inc. One-Tailed Z Test Finding Critical Z What is Z given  =.025?  =.025  

© 1998 Prentice-Hall, Inc. One-Tailed Z Test Finding Critical Z Standardized Normal Probability Table (Portion) What is Z given  =.025?  =.025  

© 1998 Prentice-Hall, Inc. One-Tailed Z Test Finding Critical Z Standardized Normal Probability Table (Portion) What is Z given  =.025?  =.025   

© 1998 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 36 boxes showed  X = The company has specified  to be 25 grams. Test at the.05 level. 368 gm.

© 1998 Prentice-Hall, Inc. One-Tailed Z Test Solution H 0 : H a :  = n = Critical Value(s): Test Statistic: Decision:Conclusion:

© 1998 Prentice-Hall, Inc. One-Tailed Z Test Solution H 0 :  = 368 H a :  > 368  = n = Critical Value(s): Test Statistic: Decision:Conclusion:

© 1998 Prentice-Hall, Inc. One-Tailed Z Test Solution H 0 :  = 368 H a :  > 368  =.05 n = 36 Critical Value(s): Test Statistic: Decision:Conclusion:

© 1998 Prentice-Hall, Inc. One-Tailed Z Test Solution H 0 :  = 368 H a :  > 368  =.05 n = 36 Critical Value(s): Test Statistic: Decision:Conclusion:

© 1998 Prentice-Hall, Inc. One-Tailed Z Test Solution H 0 :  = 368 H a :  > 368  =.05 n = 25 Critical Value(s): Test Statistic: Decision:Conclusion:

© 1998 Prentice-Hall, Inc. One-Tailed Z Test Solution H 0 :  = 368 H a :  > 368  =.05 n = 25 Critical Value(s): Test Statistic: Decision:Conclusion: Reject at  =.05

© 1998 Prentice-Hall, Inc. One-Tailed Z Test Solution H 0 :  = 368 H a :  > 368  =.05 n = 25 Critical Value(s): Test Statistic: Decision:Conclusion: Reject at  =.05 Evidence average is more than 368

© 1998 Prentice-Hall, Inc. One-Tailed Z Test Thinking Challenge You’re an analyst for Ford. You want to find out if the average miles per gallon of Escorts is at least 32 mpg. Similar models have a standard deviation of 3.8 mpg. You take a sample of 60 Escorts & compute a sample mean of 30.7 mpg. At the.01 level, is there evidence that the miles per gallon is at least 32? AloneGroupClass

© 1998 Prentice-Hall, Inc. One-Tailed Z Test Solution* H 0 : H a :  = n = Critical Value(s): Test Statistic: Decision:Conclusion: Decision:Conclusion:

© 1998 Prentice-Hall, Inc. One-Tailed Z Test Solution* H 0 :  = 32 H a :  < 32  = n = Critical Value(s): Test Statistic: Decision:Conclusion: Decision:Conclusion:

© 1998 Prentice-Hall, Inc. One-Tailed Z Test Solution* H 0 :  = 32 H a :  < 32  =.01 n = 60 Critical Value(s): Test Statistic: Decision:Conclusion: Decision:Conclusion:

© 1998 Prentice-Hall, Inc. One-Tailed Z Test Solution* H 0 :  = 32 H a :  < 32  =.01 n = 60 Critical Value(s): Test Statistic: Decision:Conclusion: Decision:Conclusion:

© 1998 Prentice-Hall, Inc. One-Tailed Z Test Solution* H 0 :  = 32 H a :  < 32  =.01 n = 60 Critical Value(s): Test Statistic: Decision:Conclusion: Decision:Conclusion:

© 1998 Prentice-Hall, Inc. One-Tailed Z Test Solution* H 0 :  = 32 H a :  < 32  =.01 n = 60 Critical Value(s): Test Statistic: Decision:Conclusion: Decision:Conclusion: Reject at  =.01

© 1998 Prentice-Hall, Inc. One-Tailed Z Test Solution* H 0 :  = 32 H a :  < 32  =.01 n = 60 Critical Value(s): Test Statistic: Decision:Conclusion: Decision:Conclusion: Reject at  =.01 There is evidence average is less than 32

© 1998 Prentice-Hall, Inc. Observed Significance Levels: p -Values

© 1998 Prentice-Hall, Inc. p -Value 1.Probability of obtaining a test statistic more extreme (  or  than the actual sample value given H 0 is true 2.Called observed level of significance Smallest value of  H 0 can be rejected Smallest value of  H 0 can be rejected 3.Used to make rejection 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

© 1998 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 36 boxes showed  X = The company has specified  to be 25 grams. Find the p-value. 368 gm.

© 1998 Prentice-Hall, Inc. Two-Tailed Z Test p -Value Solution

© 1998 Prentice-Hall, Inc. Two-Tailed Z Test p -Value Solution Z value of sample statistic (observed) 

© 1998 Prentice-Hall, Inc. Two-Tailed Z Test p -Value Solution Z value of sample statistic (observed)  p-value = P(Z  or Z  1.80)

© 1998 Prentice-Hall, Inc. Two-Tailed Z Test p -Value Solution Z value of sample statistic (observed)  p-value = P(Z  or Z  1.80)

© 1998 Prentice-Hall, Inc. Two-Tailed Z Test p -Value Solution Z value of sample statistic (observed) From Z table: lookup  p-value = P(Z  or Z  1.80)

© 1998 Prentice-Hall, Inc. Two-Tailed Z Test p -Value Solution Z value of sample statistic (observed) From Z table: lookup   p-value = P(Z  or Z  1.80)

© 1998 Prentice-Hall, Inc. Two-Tailed Z Test p -Value Solution p-value = P(Z  or Z  1.80) =.0718 Z value of sample statistic From Z table: lookup  

© 1998 Prentice-Hall, Inc. Two-Tailed Z Test p -Value Solution 1/2 p-value = /2  =.025

© 1998 Prentice-Hall, Inc. Two-Tailed Z Test p -Value Solution 1/2 p-value = /2  =.025 (p-value =.0718)  (  =.05). Do not reject.

© 1998 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 36 boxes showed  X = The company has specified  to be 25 grams. Find the p-value. 368 gm.

© 1998 Prentice-Hall, Inc. One-Tailed Z Test p -Value Solution

© 1998 Prentice-Hall, Inc. One-Tailed Z Test p -Value Solution Z Z 0 0

© 1998 Prentice-Hall, Inc. One-Tailed Z Test p -Value Solution Use alternative hypothesis to find direction Z Z 0 0 p-value 

© 1998 Prentice-Hall, Inc.   One-Tailed Z Test p -Value Solution Use alternative hypothesis to find direction Z value of sample statistic 1.80

© 1998 Prentice-Hall, Inc.   One-Tailed Z Test p -Value Solution Use alternative hypothesis to find direction Z value of sample statistic p-value is P(Z  1.80)

© 1998 Prentice-Hall, Inc. One-Tailed Z Test p -Value Solution Use alternative hypothesis to find direction p-value is P(Z  1.80) Z value of sample statistic From Z table: lookup   

© 1998 Prentice-Hall, Inc One-Tailed Z Test p -Value Solution Use alternative hypothesis to find direction p-value is P(Z  1.80) Z value of sample statistic From Z table: lookup 1.80    

© 1998 Prentice-Hall, Inc One-Tailed Z Test p -Value Solution Z value of sample statistic From Z table: lookup 1.80 Use alternative hypothesis to find direction   p-value is P(Z  1.80) =.0359  

© 1998 Prentice-Hall, Inc. Z Z p-value One-Tailed Z Test p -Value Solution =.0359 =.0359

© 1998 Prentice-Hall, Inc. Z Z p-value One-Tailed Z Test p -Value Solution =.0359 =.0359 Reject  =.05

© 1998 Prentice-Hall, Inc. Z Z p-value One-Tailed Z Test p -Value Solution =.0359 =.0359 Reject  =.05

© 1998 Prentice-Hall, Inc. (p-value =.0359)  (  =.05). Reject. One-Tailed Z Test p -Value Solution =.0359 =.0359 Reject  =.05

© 1998 Prentice-Hall, Inc. p -Value Thinking Challenge You’re an analyst for Ford. You want to find out if the average miles per gallon of Escorts is at least 32 mpg. Similar models have a standard deviation of 3.8 mpg. You take a sample of 60 Escorts & compute a sample mean of 30.7 mpg. What is the value of the observed level of significance (p-value)? AloneGroupClass

© 1998 Prentice-Hall, Inc. p -Value Solution* Z Z 0 0

© 1998 Prentice-Hall, Inc. p -Value Solution* Use alternative hypothesis to find direction 

© 1998 Prentice-Hall, Inc. p -Value Solution* Z value of sample statistic Use alternative hypothesis to find direction  

© 1998 Prentice-Hall, Inc. p -Value Solution* Z value of sample statistic From Z table: lookup Use alternative hypothesis to find direction  

© 1998 Prentice-Hall, Inc. p -Value Solution* Z value of sample statistic From Z table: lookup Use alternative hypothesis to find direction   

© 1998 Prentice-Hall, Inc. p -Value Solution* Z value of sample statistic From Z table: lookup Use alternative hypothesis to find direction    p-value = P(Z  -2.65) =.004. p-value < (  =.01). Reject H 0.

© 1998 Prentice-Hall, Inc. Two-Tailed t Test of Mean (Small Sample)

© 1998 Prentice-Hall, Inc. One Population Tests

© 1998 Prentice-Hall, Inc. t Test for Mean (Small Sample) 1.Assumptions Sample size is less than 30 (n < 30) Sample size is less than 30 (n < 30) Population is normally distributed Population is normally distributed Population standard deviation is unknown Population standard deviation is unknown

© 1998 Prentice-Hall, Inc. t Test for Mean (Small Sample) 1.Assumptions Sample size is less than 30 (n < 30) Sample size is less than 30 (n < 30) Population is normally distributed Population is normally distributed Population standard deviation is unknown Population standard deviation is unknown 3.T test statistic

© 1998 Prentice-Hall, Inc. Two-Tailed t Test Finding Critical t Values

© 1998 Prentice-Hall, Inc. Two-Tailed t Test Finding Critical t Values Given: n = 3;  =.10

© 1998 Prentice-Hall, Inc. Two-Tailed t Test Finding Critical t Values  /2 =.05  Given: n = 3;  =.10

© 1998 Prentice-Hall, Inc. Two-Tailed t Test Finding Critical t Values  /2 =.05   Given: n = 3;  =.10 df = n - 1 = 2

© 1998 Prentice-Hall, Inc. Two-Tailed t Test Finding Critical t Values Critical Values of t Table (Portion)  /2 =.05    Given: n = 3;  =.10 df = n - 1 = 2

© 1998 Prentice-Hall, Inc. Two-Tailed t Test Finding Critical t Values Critical Values of t Table (Portion)  /2 =.05     Given: n = 3;  =.10 df = n - 1 = 2

© 1998 Prentice-Hall, Inc. Two-Tailed t Test Example Does an average box of cereal contain 368 grams of cereal? A random sample of 25 boxes had a mean of & a standard deviation of 12 grams. Test at the.05 level. 368 gm.

© 1998 Prentice-Hall, Inc. Two-Tailed t Test Solution H 0 : H a :  = df = Critical Value(s): Test Statistic: Decision:Conclusion:

© 1998 Prentice-Hall, Inc. Two-Tailed t Test Solution H 0 :  = 368 H a :   368  = df = Critical Value(s): Test Statistic: Decision:Conclusion:

© 1998 Prentice-Hall, Inc. Two-Tailed t Test Solution H 0 :  = 368 H a :   368  =.05 df = = 24 Critical Value(s): Test Statistic: Decision:Conclusion:

© 1998 Prentice-Hall, Inc. Two-Tailed t Test Solution H 0 :  = 368 H a :   368  =.05 df = = 24 Critical Value(s): Test Statistic: Decision:Conclusion:

© 1998 Prentice-Hall, Inc. Two-Tailed t Test Solution H 0 :  = 368 H a :   368  =.05 df = = 24 Critical Value(s): Test Statistic: Decision:Conclusion:

© 1998 Prentice-Hall, Inc. Two-Tailed t Test Solution H 0 :  = 368 H a :   368  =.05 df = = 24 Critical Value(s): Test Statistic: Decision:Conclusion: Do not reject at  =.05

© 1998 Prentice-Hall, Inc. Two-Tailed t Test Solution H 0 :  = 368 H a :   368  =.05 df = = 24 Critical Value(s): Test Statistic: Decision:Conclusion: Do not reject at  =.05 There is no evidence pop. average is not 368

© 1998 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 16 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. AloneGroupClass

© 1998 Prentice-Hall, Inc. Two-Tailed t Test Solution* H 0 : H a :   df  Critical Value(s): Test Statistic: Decision:Conclusion:

© 1998 Prentice-Hall, Inc. Two-Tailed t Test Solution* H 0 :  = 3.25 H a :   3.25   df  Critical Value(s): Test Statistic: Decision:Conclusion:

© 1998 Prentice-Hall, Inc. Two-Tailed t Test Solution* H 0 :  = 3.25 H a :   3.25  .01 df  = 15 Critical Value(s): Test Statistic: Decision:Conclusion:

© 1998 Prentice-Hall, Inc. Two-Tailed t Test Solution* H 0 :  = 3.25 H a :   3.25  .01 df  = 15 Critical Value(s): Test Statistic: Decision:Conclusion:

© 1998 Prentice-Hall, Inc. Two-Tailed t Test Solution* H 0 :  = 3.25 H a :   3.25  .01 df  = 15 Critical Value(s): Test Statistic: Decision:Conclusion:

© 1998 Prentice-Hall, Inc. Two-Tailed t Test Solution* H 0 :  = 3.25 H a :   3.25  .01 df  = 15 Critical Value(s): Test Statistic: Decision:Conclusion: Do not reject at  =.01

© 1998 Prentice-Hall, Inc. Two-Tailed t Test Solution* H 0 :  = 3.25 H a :   3.25  .01 df  = 15 Critical Value(s): Test Statistic: Decision:Conclusion: Do not reject at  =.01 There is no evidence average is not 3.25

© 1998 Prentice-Hall, Inc. One-Tailed t Test of Mean (Small Sample)

© 1998 Prentice-Hall, Inc. One-Tailed t Test Example Is the average capacity of batteries at least 140 ampere-hours? A random sample of 20 batteries had a mean of & a standard deviation of Assume a normal distribution. Test at the.05 level.

© 1998 Prentice-Hall, Inc. One-Tailed t Test Solution H 0 : H a :  = df = Critical Value(s): Test Statistic: Decision:Conclusion:

© 1998 Prentice-Hall, Inc. One-Tailed t Test Solution H 0 :  = 140 H a :  < 140  = df = Critical Value(s): Test Statistic: Decision:Conclusion:

© 1998 Prentice-Hall, Inc. One-Tailed t Test Solution H 0 :  = 140 H a :  < 140  =.05 df = = 19 Critical Value(s): Test Statistic: Decision:Conclusion:

© 1998 Prentice-Hall, Inc. One-Tailed t Test Solution H 0 :  = 140 H a :  < 140  =.05 df = = 19 Critical Value(s): Test Statistic: Decision:Conclusion:

© 1998 Prentice-Hall, Inc. One-Tailed t Test Solution H 0 :  = 140 H a :  < 140  =.05 df = = 19 Critical Value(s): Test Statistic: Decision:Conclusion:

© 1998 Prentice-Hall, Inc. One-Tailed t Test Solution H 0 :  = 140 H a :  < 140  =.05 df = = 19 Critical Value(s): Test Statistic: Decision:Conclusion: Reject at  =.05

© 1998 Prentice-Hall, Inc. One-Tailed t Test Solution H 0 :  = 140 H a :  < 140  =.05 df = = 19 Critical Value(s): Test Statistic: Decision:Conclusion: Reject at  =.05 There is evidence pop. average is less than 140

© 1998 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)? AloneGroupClass

© 1998 Prentice-Hall, Inc. One-Tailed t Test Solution* H 0 : H a :  = df = Critical Value(s): Test Statistic: Decision:Conclusion:

© 1998 Prentice-Hall, Inc. One-Tailed t Test Solution* H 0 :  = 5 H a :  > 5  = df = Critical Value(s): Test Statistic: Decision:Conclusion:

© 1998 Prentice-Hall, Inc. One-Tailed t Test Solution* H 0 :  = 5 H a :  > 5  =.05 df = = 9 Critical Value(s): Test Statistic: Decision:Conclusion:

© 1998 Prentice-Hall, Inc. One-Tailed t Test Solution* H 0 :  = 5 H a :  > 5  =.05 df = = 9 Critical Value(s): Test Statistic: Decision:Conclusion:

© 1998 Prentice-Hall, Inc. One-Tailed t Test Solution* H 0 :  = 5 H a :  > 5  =.05 df = = 9 Critical Value(s): Test Statistic: Decision:Conclusion:

© 1998 Prentice-Hall, Inc. One-Tailed t Test Solution* H 0 :  = 5 H a :  > 5  =.05 df = = 9 Critical Value(s): Test Statistic: Decision:Conclusion: Do not reject at  =.05

© 1998 Prentice-Hall, Inc. One-Tailed t Test Solution* H 0 :  = 5 H a :  > 5  =.05 df = = 9 Critical Value(s): Test Statistic: Decision:Conclusion: Do not reject at  =.05 There is no evidence average is more than 5

© 1998 Prentice-Hall, Inc. Z Test of Proportion

© 1998 Prentice-Hall, Inc. One Population Tests One Population Z Test (1 & 2 tail) t Test (1 & 2 tail) Large Sample Z Test (1 & 2 tail) MeanProportion Small Sample

© 1998 Prentice-Hall, Inc. One-Sample Z Test for Proportion

© 1998 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

© 1998 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

© 1998 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.

© 1998 Prentice-Hall, Inc. One-Proportion Z Test Solution H 0 : H a :  = n = Critical Value(s): Test Statistic: Decision:Conclusion:

© 1998 Prentice-Hall, Inc. One-Proportion Z Test Solution H 0 : p =.10 H a : p <.10  = n = Critical Value(s): Test Statistic: Decision:Conclusion:

© 1998 Prentice-Hall, Inc. One-Proportion Z Test Solution H 0 : p = .10 H a : p <.10  =.05 n = 200 Critical Value(s): Test Statistic: Decision:Conclusion:

© 1998 Prentice-Hall, Inc. One-Proportion Z Test Solution H 0 : p =.10 H a : p <.10  =.05 n = 200 Critical Value(s): Test Statistic: Decision:Conclusion:

© 1998 Prentice-Hall, Inc. One-Proportion Z Test Solution H 0 : p =.10 H a : p <.10  =.05 n = 200 Critical Value(s): Test Statistic: Decision:Conclusion:

© 1998 Prentice-Hall, Inc. One-Proportion Z Test Solution H 0 : p =.10 H a : p <.10  =.05 n = 200 Critical Value(s): Test Statistic: Decision:Conclusion: Reject at  =.05

© 1998 Prentice-Hall, Inc. One-Proportion Z Test Solution H 0 : p =.10 H a : p <.10  =.05 n = 200 Critical Value(s): Test Statistic: Decision:Conclusion: Reject at  =.05 There is evidence new system < 10% defective

© 1998 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? AloneGroupClass

© 1998 Prentice-Hall, Inc. One-Proportion Z Test Solution* H 0 : H a :  = n = Critical Value(s): Test Statistic: Decision:Conclusion:

© 1998 Prentice-Hall, Inc. One-Proportion Z Test Solution* H 0 : p =.04 H a : p .04  = n = Critical Value(s): Test Statistic: Decision:Conclusion:

© 1998 Prentice-Hall, Inc. One-Proportion Z Test Solution* H 0 : p =.04 H a : p .04  =.05 n = 500 Critical Value(s): Test Statistic: Decision:Conclusion:

© 1998 Prentice-Hall, Inc. One-Proportion Z Test Solution* H 0 : p =.04 H a : p .04  =.05 n = 500 Critical Value(s): Test Statistic: Decision:Conclusion:

© 1998 Prentice-Hall, Inc. One-Proportion Z Test Solution* H 0 : p =.04 H a : p .04  =.05 n = 500 Critical Value(s): Test Statistic: Decision:Conclusion:

© 1998 Prentice-Hall, Inc. One-Proportion Z Test Solution* H 0 : p =.04 H a : p .04  =.05 n = 500 Critical Value(s): Test Statistic: Decision:Conclusion: Do not reject at  =.05

© 1998 Prentice-Hall, Inc. One-Proportion Z Test Solution* H 0 : p =.04 H a : p .04  =.05 n = 500 Critical Value(s): Test Statistic: Decision:Conclusion: Do not reject at  =.05 There is evidence proportion is still 4%

© 1998 Prentice-Hall, Inc. Conclusion 1.Distinguished types of hypotheses 2.Described hypothesis testing process 3.Explained p-value concept 4.Solved hypothesis testing problems based on a single sample

© 1998 Prentice-Hall, Inc. This Class... 1.What was the most important thing you learned in class today? 2.What do you still have questions about? 3.How can today’s class be improved? Please take a moment to answer the following questions in writing:

End of Chapter Any blank slides that follow are blank intentionally.