Hypothesis Testing Variance known?

Sampling Distribution n Over-the-counter stock selling prices calculate average price of all stocks listed [  ]calculate average price of all stocks listed [  ] n take a sample of 25 stocks and record price calculate average price of the 25 stocks [x-bar]calculate average price of the 25 stocks [x-bar] n take all possible samples of size 25 would all x-bars be equal?would all x-bars be equal? n average all the possible x-bars …equals 

Levine, Prentice-Hall Sampling Distribution 20 H0H0

Levine, Prentice-Hall Sampling Distribution It is unlikely that we would get a sample mean of this value H0H0

Levine, Prentice-Hall Sampling Distribution It is unlikely that we would get a sample mean of this value if in fact this were the population mean 20 H0H0

Levine, Prentice-Hall 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  = H0H0

Levine, Prentice-Hall Null Hypothesis n What is tested Always has equality sign: ,  or  Always has equality sign: ,  or  n Designated H 0 Example ………... H 0 :   3 Example ………... H 0 :   3

Levine, Prentice-Hall Alternative Hypothesis Opposite of null hypothesis Opposite of null hypothesis Always has inequality sign: , , or  Always has inequality sign: , , or  n Designated H 1 n Example H 1 :  < 3 H 1 :  < 3

Levine, Prentice-Hall Decision Reject null hypothesis Reject null hypothesis n Retain, or, fail to reject, null hypothesis n Do not use the term “accept”

Levine, Prentice-Hall p-value Probability of obtaining a test statistic more extreme (  or  than actual sample value given H 0 is true Probability of obtaining a test statistic more extreme (  or  than actual sample value given H 0 is true n Called observed level of significance Smallest value of  H 0 can be rejected Smallest value of  H 0 can be rejected n Used to make rejection decision If p-value  , reject H 0 If p-value  , reject H 0

Levine, Prentice-Hall Level of Significance n Defines unlikely values of sample statistic if null hypothesis is true Called rejection region of sampling distribution Called rejection region of sampling distribution Designated  (alpha) Designated  (alpha) Typical values are.01,.05,.10 Typical values are.01,.05,.10 n Selected by researcher at start

Levine, Prentice-Hall Rejection Region (one- tail test) Sampling Distribution 1 -  Level of Confidence

Levine, Prentice-Hall Rejection Region (one- tail test) Sampling Distribution 1 -  Level of Confidence Observed sample statistic

Levine, Prentice-Hall Rejection Region (one- tail test) Sampling Distribution 1 -  Level of Confidence Observed sample statistic

Levine, Prentice-Hall Rejection Regions (two- tailed test) Sampling Distribution 1 -  Level of Confidence

Levine, Prentice-Hall Rejection Regions (two- tailed test) Sampling Distribution 1 -  Level of Confidence Observed sample statistic

Levine, Prentice-Hall Rejection Regions (two- tailed test) Sampling Distribution 1 -  Level of Confidence Observed sample statistic

Levine, Prentice-Hall Rejection Regions (two- tailed test) Sampling Distribution 1 -  Level of Confidence Observed sample statistic

Levine, Prentice-Hall Risk of Errors in Making Decision n 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 n 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 [ 

Levine, Prentice-Hall Decision Results H 0 : Innocent

Levine, Prentice-Hall Hypothesis Testing n State H 0 n State H 1 Choose  Choose  n Choose n n Choose test

Levine, Prentice-Hall Hypothesis Testing 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

Levine, Prentice-Hall Two-tailed z-test Does an average box of cereal contain 368 grams of cereal? A random sample of 25 boxes has an average weight = grams. The company has specified  to be 15 grams. Test at the.05 level. 368 gm.

Levine, Prentice-Hall Two-tailed z-test H 0 : H 1 :   n  Critical Value(s): Test Statistic: Decision:Conclusion:

Levine, Prentice-Hall Two-tailed z-test H 0 :  = 368 H 1 :   368   n  Critical Value(s): Test Statistic: Decision:Conclusion:

Levine, Prentice-Hall Two-tailed z-test H 0 :  = 368 H 1 :   368  .05 n  25 Critical Value(s): Test Statistic: Decision:Conclusion:

Levine, Prentice-Hall Two-tailed z-test H 0 :  = 368 H 1 :   368  .05 n  25 Critical Value(s): Test Statistic: Decision:Conclusion:

Levine, Prentice-Hall Two-tailed z-test H 0 :  = 368 H 1 :   368  .05 n  25 Critical Value(s): Test Statistic: Decision:Conclusion:

Levine, Prentice-Hall Two-tailed z-test H 0 :  = 368 H 1 :   368  .05 n  25 Critical Value(s): Test Statistic: Decision:Conclusion: Do not reject at  =.05

Levine, Prentice-Hall Two-tailed z-test H 0 :  = 368 H 1 :   368  .05 n  25 Critical Value(s): Test Statistic: Decision:Conclusion: Do not reject at  =.05 No evidence average is not 368

Levine, Prentice-Hall Two-tailed z-test [p-value]] Z value of sample statistic 

Levine, Prentice-Hall Two-tailed z-test [p-value] p-value is P(z  or z  1.50) Z value of sample statistic 

Levine, Prentice-Hall Two-tailed z-test [p-value] p-value is P(z  or z  1.50) Z value of sample statistic 

Levine, Prentice-Hall Two-tailed z-test [p-value] p-value is P(z  or z  1.50) Z value of sample statistic From Z table: lookup 

Levine, Prentice-Hall Two-tailed z-test [p-value] p-value is P(z  or z  1.50) Z value of sample statistic From Z table: lookup  

Levine, Prentice-Hall Two-tailed z-test [p-value] p-value is P(z  or z  1.50) =.1336 Z value of sample statistic From Z table: lookup  

Levine, Prentice-Hall Two-tailed z-test [p-value] 1/2 p-value = /2  =.025

Levine, Prentice-Hall Two-tailed z-test [p-value] (p-Value =.1336)  (  =.05) Do not reject. 1/2 p-Value = /2  =.025 Test statistic is in ‘Do not reject’ region

Levine, Prentice-Hall Two-tailed z-test (  known) challenge You are 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?

Levine, Prentice-Hall solution template (  known) H 0 : H 1 :  = n = Critical Value(s): Test Statistic: Decision: Conclusion:

Levine, Prentice-Hall Two-tailed z-test (  known) H 0 : H 1 :  = n = Critical Value(s): Test Statistic: Decision: Conclusion:

Levine, Prentice-Hall Two-tailed z-test (  known) H 0 :  = 70 H 1 :   70  = n = Critical Value(s): Test Statistic: Decision: Conclusion:

Levine, Prentice-Hall Two-tailed z-test (  known) H 0 :  = 70 H 1 :   70  =.05 n = 36 Critical Value(s): Test Statistic: Decision: Conclusion:

Levine, Prentice-Hall Two-tailed z-test (  known) H 0 :  = 70 H 1 :   70  =.05 n = 36 Critical Value(s): Test Statistic: Decision: Conclusion:

Levine, Prentice-Hall Two-tailed z-test (  known) H 0 :  = 70 H 1 :   70  =.05 n = 36 Critical Value(s): Test Statistic: Decision: Conclusion:

Levine, Prentice-Hall Two-tailed z-test (  known) H 0 :  = 70 H 1 :   70  =.05 n = 36 Critical Value(s): Test Statistic: Decision: Conclusion: Do not reject at  =.05

Levine, Prentice-Hall Two-tailed z-test (  known) H 0 :  = 70 H 1 :   70  =.05 n = 36 Critical Value(s): Test Statistic: Decision: Conclusion: Do not reject at  =.05 No evidence average is not 70

Levine, Prentice-Hall One-tailed z-test (  known) n Assumptions Population is normally distributed Population is normally distributed If not normal, can be approximated by normal distribution for large samples If not normal, can be approximated by normal distribution for large samples

Levine, Prentice-Hall One-tailed z-test (  known) n Assumptions Population is normally distributed Population is normally distributed If not normal, can be approximated by normal distribution for large samples If not normal, can be approximated by normal distribution for large samples Null hypothesis has  or  sign only Null hypothesis has  or  sign only

Levine, Prentice-Hall One-tailed z-test (  known) n Assumptions Population is normally distributed Population is normally distributed If not normal, can be approximated by normal distribution for large samples If not normal, can be approximated by normal distribution for large samples Null hypothesis has  or  sign only Null hypothesis has  or  sign only n Z-test statistic

Levine, Prentice-Hall One-tailed z-test (  known) H 0 :  0 H 1 :  < 0 Must be significantly below 

Levine, Prentice-Hall One-tailed z-test (  known) H 0 :  0 H 1 :  < 0H 0 :  0 H 1 :  > 0 Must be significantly below  Small values satisfy H 0. Do not reject!

Levine, Prentice-Hall One-tailed z-test (  known) What is “z” given  =.025?  =.025 

Levine, Prentice-Hall One-tailed z-test (  known) What Is Z given  =.025?  =.025  

Levine, Prentice-Hall One-tailed z-test (  known) Standardized Normal Probability Table (Portion) What is “z” given  =.025?  =.025  

Levine, Prentice-Hall One-tailed z-test (  known) Standardized Normal Probability Table (Portion) What Is Z given  =.025?  =.025   

Levine, Prentice-Hall One-tailed z-test (  known) 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 15 grams. Test at the.05 level. 368 gm.

Levine, Prentice-Hall One-tailed z-test (  known) H 0 : H 1 :  = n = Critical Value(s): Test Statistic: Decision: Conclusion:

Levine, Prentice-Hall One-tailed z-test (  known) H 0 :   368 H 1 :  > 368  = n = Critical Value(s): Test Statistic: Decision: Conclusion:

Levine, Prentice-Hall One-tailed z-test (  known) H 0 :   368 H 1 :  > 368  =.05 n = 25 Critical Value(s): Test Statistic: Decision: Conclusion:

Levine, Prentice-Hall One-tailed z-test (  known) H 0 :   368 H 1 :  > 368  =.05 n = 25 Critical Value(s): Test Statistic: Decision: Conclusion:

Levine, Prentice-Hall One-tailed z-test (  known) H 0 :   368 H 1 :  > 368  =.05 n = 25 Critical Value(s): Test Statistic: Decision: Conclusion:

Levine, Prentice-Hall One-tailed z-test (  known) H 0 :   368 H 1 :  > 368  =.05 n = 25 Critical Value(s): Test Statistic: Decision: Conclusion: Do not reject at  =.05

Levine, Prentice-Hall One-tailed z-test (  known) H 0 :   368 H 1 :  > 368  =.05 n = 25 Critical Value(s): Test Statistic: Decision: Conclusion: Do not reject at  =.05 No evidence average is more than 368

Levine, Prentice-Hall One-tailed z-test (  known) p-value Solution Z value of sample statistic Use alternative hypothesis to find direction  

Levine, Prentice-Hall One-tailed z-test (  known) p-value Z value of sample statistic Use alternative hypothesis to find direction  

Levine, Prentice-Hall One-tailed z-test (  known) p-value Z value of sample statistic From Z table: lookup Use alternative hypothesis to find direction   

Levine, Prentice-Hall One-tailed z-test (  known) p-value Z value of sample statistic From Z table: lookup Use alternative hypothesis to find direction    

Levine, Prentice-Hall One-tailed z-test (  known) p-value Z value of sample statistic From Z table: lookup Use alternative hypothesis to find direction     p-value =.0668

Levine, Prentice-Hall One-tailed z-test (  known) p-value p-value =.0668  =.05

Levine, Prentice-Hall One-tailed z-test (  known) p-value (p-value =.0668)  (  =.05). Do not reject. p-Value =.0668  =.05 Test statistic is in ‘Fail to reject’ region

Levine, Prentice-Hall p-value 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)?

Levine, Prentice-Hall p-value Z value of sample statistic From Z table: lookup Use alternative hypothesis to find direction    p-value =.004 p-value <  (  =.01) Reject H 0.

Levine, Prentice-Hall p-value Probability of obtaining a test statistic more extreme (  or  than actual sample value given H 0 is true Probability of obtaining a test statistic more extreme (  or  than actual sample value given H 0 is true n Called observed level of significance Smallest value of  H 0 can be rejected Smallest value of  H 0 can be rejected n Used to make rejection decision If p-value  , reject H 0 If p-value  , reject H 0

Levine, Prentice-Hall One-tailed t-test (  unknown) Does an average box of cereal contain less than the 368 grams indicated on the package? A random sample of 25 boxes showed  X = and s=15. Test at the.05 level. 368 gr.

Levine, Prentice-Hall H 0 : H 1 :  = n = Critical Value(s): Test Statistic: Decision: Conclusion: One-tailed t-test (  unknown)

Levine, Prentice-Hall H 0 :   368 H 1 :  < 368  = n = Critical Value(s): Test Statistic: Decision: Conclusion: One-tailed t-test (  unknown)

Levine, Prentice-Hall H 0 :   368 H 1 :  < 368  =.05 n = 25, d.f. = 24 Critical Value(s): Test Statistic: Decision: Conclusion: One-tailed t-test (  unknown)

Levine, Prentice-Hall H 0 :   368 H 1 :  < 368  =.05 n = 25, d.f. = 24 Critical Value(s): Test Statistic: Decision: Conclusion: One-tailed t-test (  unknown)

Levine, Prentice-Hall H 0 :   368 H 1 :  < 368  =.05 n = 25, d.f. = 24 Critical Value(s): Test Statistic: Decision: Conclusion: One-tailed t-test (  unknown)

Levine, Prentice-Hall H 0 :   368 H 1 :  < 368  =.05 n = 25, d.f. = 24 Critical Value(s): Test Statistic: Decision: Conclusion: Do not reject at  =.05 One-tailed t-test (  unknown)

Levine, Prentice-Hall H 0 :   368 H 1 :  < 368  =.05 n = 25, d.f. = 24 Critical Value(s): Test Statistic: Decision: Conclusion: Do not reject at  =.05 No evidence average is less than 368 One-tailed t-test (  unknown)

Levine, Prentice-Hall One-tailed t-test (  unknown) p-value Solution t value of sample statistic   Use alternative hypothesis to find direction

Levine, Prentice-Hall One-tailed t-test (  unknown) p-value  Use alternative hypothesis to find direction t value of sample statistic   From t table: lookup for 24 d.f. P-value = 0.075

Levine, Prentice-Hall p-value =.075  =.05 One-tailed t-test (  unknown) p-value

Levine, Prentice-Hall p-value =.075  =.05 One-tailed t-test (  unknown) p-value Test statistic is in ‘Fail to reject’ region (p-value =.075)  (  =.05). Do not reject. Reject

Questions?

ANOVA