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MAT 214 Lab 11: Decision making FALL 2008 In this lab we study how to use statistics to make a decision, as well as calculate the probabilities of making a correct and incorrect decision. We will use Minitab as an aid in making many decisions and tracking the quality of those decisions. I. The problem: Decide whether the mean of a population, , is greater than 5 or not greater than 5. Procedure for making a decision: Take a random sample of size 100 and compute X . If X 7, decide that the mean of the population, , is not greater than 5. If X > 7, decide that the mean of the population, , is greater than 5. II. If Statement Minitab does not have an “if” statement at the command level, so to execute an “if” statement we must tell Minitab by using a special file called a Macro. The file IfStatement.mac contains the instructions Minitab needs to implement the “if” statement at the command level. Copy this file to your p: drive. When the following line is put into the command line editor %p:\IfStatement.mac 'xBar' 7 'The Decision' it tells Minitab to make ‘The Decision’ equal 0 if xBar is less than or equal to 7, and make ‘The Decision’ equal 1 if xBar is greater than 7. [Technically speaking, Minitab executes the following instruction: If ‘xBar’ 7 then ‘The Decision’ = 0 else ‘The Decision’ = 1 end if ] Note, a value of ‘Decision’ equal to 1 means we decide > 5 and a value of ‘Decision’ equal to 0 means we decided ≤ 5. MAT 214 Lab 11: Decision making III. FALL 2008 Select a random sample of size 100 from a normal distribution. a. Open Notepad Click Start in the lower left hand corner of the screen Click Programs Click Accessories Click Notepad b. Now we’ll type the Minitab commands into Notepad that instruct Minitab to generate a random sample of size n = 100 from a normal distribution with mean Mu (we’ll specify Mu later) and standard deviation 12, compute the sample means, and then execute the if statement (decision rule). Then it saves ‘The Decision’ in the column named Decisions. Type the following instructions into Notepad. Random 100 'Data'; Normal 'Mu', 12. Let 'xBar' = mean('Data') %p:\IfStatement.mac 'xBar' 7 'The Decision' Let 'Decisions' (k1) = 'The Decision' Let k1 = k1 + 1 Save the contents of your Notepad document to the file p:\Decision.txt. c. Let’s let our mean be 5 (i.e ‘Mu’ = 5), and tell Minitab to select 150 random samples of size 100, and for each of the random samples of size 100 execute Decision.txt. Go to Minitab Click Edit Click Command Line Editor Type the following lines in the Command Line Editor Window Erase C1, C2 Name C1 'Data' Name C2 'Decisions' Name k2 'xBar' Name k3 'Mu' Name k4 'The Decision' Let k1 = 1 Let 'Mu' = 5 Execute "p:\Decision.txt" 150 Click Submit Commands MAT 214 Lab 11: Decision making FALL 2008 d. Make a frequency table and determine how many times the decision rule decided that the mean was ≤ 5 or > 5 (i.e. how many times “The Decision” was 0 or 1). Click Stat Click Tally Double Click C2 (Decisions) Click OK Note the results. Recall the number of 0’s is the number of times the correct decision was made (i.e. the mean of the population is ≤ 5). Record the proportion of the samples for which the decision rule decided the mean is ≤ 5 and the proportion of the samples for which the rule decided the mean is > 5. Is this what you expected to see? IV. Repeat Part III. c and d changing the mean ‘Mu’ to 8. Summarize the results again as you did above except noting that a right decision is that the mean of the population is > 5 (i.e. ‘The Decision’ = 1). V. Compute the probability X 7 when is 5 and compare these results with III. d. Compute the probability X > 7 when is 8 and compare these results with IV. VI. Repeat Part III. c and d changing the mean to 7.1. Summarize the results again as you did above. In this case, the decision rule works as follows: If X 7, we conclude that ≤ 5, and if X > 7, we conclude that > 5. Comment on the ability of this decision rule to distinguish a mean 7.1 from a mean of 5.