M34- Hypothesis Testing 1  Department of ISM, University of Alabama, Homework Chp 11. Use equations on formula sheet. Page 417 # 5, 6, 8. (Use CI to make a decision). Page 441 # 47, 49, 50. (Use CI to make....) Chp 15, Read 633 bottom, 634 middle. Page 636 # 50 (Do part c to answer part b.) # 51. (Do part c to answer part b.) # 53, 54 (Review) # 47, 48. (Give answer for p-value (see answer to 47 ibob) in relation to the  -level.)

M34- Hypothesis Testing 2  Department of ISM, University of Alabama, Hypothesis Testing

M34- Hypothesis Testing 3  Department of ISM, University of Alabama, Lesson Objectives  Understand the “types of errors” in decision making.  Know what the  -level means.  Learn how to use “p-values” and confidence intervals for decision making.

M34- Hypothesis Testing 4  Department of ISM, University of Alabama, Court case Hypothesis:Defendant is innocent. Alternative:Defendant is guilty. Decisions: Based on the sample data. Reject Innocence “Guilty” Declare “Guilty” Person goes to jail! Do not Reject Innocence “Not Guilty” Declare “Not Guilty” Person goes free!

M34- Hypothesis Testing 5  Department of ISM, University of Alabama, Type I:Sending an innocent person to jail. Type II:Letting a guilty person go free.  =level of risk deemed reasonable for the occurrence of a Type I error. = the point of “reasonable doubt.” Types of Errors in a court case

M34- Hypothesis Testing 6  Department of ISM, University of Alabama, Types of Errors, in general Type I:Concluding that the hypothesized parameter value is wrong, but in reality it is correct. Type II:Not concluding that the hypothesized parameter value is wrong, but in reality it is incorrect.  = level of risk, chosen by the user, for allowing a Type I error to occur.  = risk for making Type II error.

 Net weight of potato chip bags should be oz.  An FDA inspector will take a random sample of 36 bags. If the net weight is too low, the chip company will be fined substantially.  From the FDA perspective, what would the Type I and Type II errors be (in words)? Potato Chip Inspection by FDA

M34- Hypothesis Testing 8  Department of ISM, University of Alabama, Type I:Penalizing the potato chip company when in reality they were NOT cheating the consumer. Type II:Not detecting that the potato chip company was cheating the consumers, when in reality they were. Potato Chips; types of errors Which is more serious, from the FDA’s perspective?

M34- Hypothesis Testing 9  Department of ISM, University of Alabama, Chose an  -level that considers the consequences of the Type I and Type II errors.  and  are inversely related; as one goes up, the other goes down, but NOT by equal amounts. Selecting an  -level

M34- Hypothesis Testing 10  Department of ISM, University of Alabama, Reject the hypothesized value if: 1.it is outside the confidence interval. 2.the p-value is less than the user specified  -level. (p-value <  -level) 3.the calculated test statistic value is in the “critical region.” Statistical Inference Methods: Three methods; each should give the same result.

M34- Hypothesis Testing 11  Department of ISM, University of Alabama, the data Decisions are based on the data. Wrong decisions are the result of chance, not mistakes.

(1-  )100% Confidence Interval Method Two tailed test ; “Is the mean something other than 40.0?” One tail test; “Is the mean something greater than 40.0?” or “Is the mean something less than 40.0?” Hypothesized mean: 40.0 Result: Each tail has the full  -level. Use only ONE tail for making a decision Desired  -level: Size of CI to use: 1 -     Result: Each tail has half of .

M34- Hypothesis Testing 13  Department of ISM, University of Alabama, p-Value Method The probability of observing a future statistic value that is as big or more extreme, in the direction(s) of interest, than the value we just observed, assuming that the hypothesized value is the correct parameter. Calculate p-value using the most appropriate distribution. Decision rule: If p-value <  -level, reject the hypothesized value.

M34- Hypothesis Testing 14  Department of ISM, University of Alabama, X = X 42.6 Hypo. mean: 40.0, p-Value: 40X X Two tailed test; “Is the mean something other than 40.0?” Upper tail test; “Is the mean something greater than 40.0?” Lower tail test; “Is the mean something less than 40.0?” p-value / 2 p-value p-value / 2 2.6

M34- Hypothesis Testing 15  Department of ISM, University of Alabama, Sample results: X = 43.0 s = 7.2 0Z Z = 43.0 – = X 43.0 Hypothesized mean: Adjust machine if it’s off in either direction. More extreme Also more extreme p-value = =.1336 than 3.0 units Problem 1, using p-Value What distribution should be used? Pick  =.05 (p-value =.1336) > (  =.05) ;  do not reject 40.0.

M34- Hypothesis Testing 16  Department of ISM, University of Alabama, Sample results: X = 43.0 s = 7.2 Hypothesized mean: Adjust machine if it’s off in either direction. Problem 1 with Confidence Interval What distribution should be used? Pick  = falls inside the C.I. ;  do not reject X ± Z  /2 nn 43.0 ± 1.96  ± 4.92 (38.08, 47.92)

M34- Hypothesis Testing 17  Department of ISM, University of Alabama, Sample results: X = 36.4 s = 7.2 0t t = 36.4 – / 4 = X 36.4 Hypothesized mean: FDA will fine company if the mean is lower. More extreme p-value =. than 3.6 units below. Problem 2, using p-Value From the t-table.... From Excel,.0320 more than.025, less than.050. What distribution should be used? (p-value =.0320) < (  -level =.05);  reject Pick  =.05

M34- Hypothesis Testing 18  Department of ISM, University of Alabama, Sample results: X = 36.4 s = 7.2 Hypothesized mean: FDA will fine company if the mean is lower. What distribution should be used? 40.0 falls outside the C.I.;  reject Pick  =.05 X ± t , 15 snsn (33.245, ) 36.4 ± ±  Problem 2 with Confidence Interval

M34- Hypothesis Testing 19  Department of ISM, University of Alabama,  Two people in different rooms.  “A” is shown one of five cards, selected randomly. “A” transmits his thoughts.  “B” selects the card he thinks is being sent to him, and records it  The process is repeated 20 times; the cards are shuffled each time. Does a person have ESP? Experiment: X = a count of the number correct. X ~ Bino(n=20,  =.20)

M34- Hypothesis Testing 20  Department of ISM, University of Alabama, X = a count of the number correct. X ~ Bino(n=20,  =.20)  = n  = 4.0 (Cannot use Normal approx.) Data: “B” got 9 out 20 correct. Does “B” do better than guessing? Hypothesized value:  =.20, p-value = P(X >= 9) = 1 – P(X <= 8) = 1 –.9900 =.0100 Use Table A.2  -level = Bino(20, 0.20) Use Binomial Dist..05 Reject.20; she does better!

M34- Hypothesis Testing 21  Department of ISM, University of Alabama, The end