5.11 Using JMP. in This Section We Solve Some of the Examples by Using JMP

5.11 Using JMP. In this section we solve some of the examples by using JMP.

Example 5.11.1 (Normal Probabilities Using JMP) Let a random variable X be distributed as normal with mean = 6 and standard deviation = 4. Determine the probability P(8.0 X 14.0).

Solution: In order to determine the probability P(8.0 X 14.0), we have to first find the probabilities P(X 8.0) and P(X 14.0). Then P(8.0 X 14.0) = P(X 14.0) – P(X 8.0). To find the probabilities P(X 8.0) and P(X 14.0) using JMP, proceed as follows:

1. From the main JMP taskbar, select File > New > Data Table, which results in an untitled data table in a separate screen. To assign a title to this table, put the cursor in the box titled “Untitled” in the left corner, right click twice, and enter the desired title.

2. Create two columns, Column 1 and Column 2, in a New Data Table.

3. Enter the values 8 and 14 in Column 1.

4. Label Column 2 as N(6,4) and right-click on N(6,4). Then select the Formula option. Another dialog box N(6,4) appears.

5. In the dialog box N(6,4) select the Probability option from the Function (grouped). Scroll down the drop-down menu and select the Normal Distribution option.

6. In the text box below Normal Distribution [x] shows up in Text box. Select Column 1 from the Table Columns, which appears within the parenthesis. After selecting Column 1, enter a comma, and then enter mean value = 6. After entering the mean, hit the ‘enter’ key, then enter a comma, and then enter the standard deviation = 4.

7. Select OK to have the normal probabilities calculated in Column 2.

8. In the Table window, the values for P(X 8.0) and P(X 14.0) show

up in the column labeled N(6,4) shown in the JMP printout as 0.6914246

and 0.97724987, respectively. Thus, P(8.0 X 14.0) = P(X 14.0) P(X 8.0)= 0.977250 0.691462 = 0.285788.

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Example 5.11.2 (Exponential Probabilities Using JMP) Let X be a random variable distributed as an exponential distribution with. Determine the following probabilities:

(a)  P() (b) P(X 5.0) (c) P(3.0 X 5.0)

Solution: In order to determine the probabilities P(), P(X 5.0), and P(3.0 X 5.0), we have to first find the probabilities P(X 3.0) and P(X 5.0). Then P() = 1- P(X 3.0), P(3.0 X 5.0) = P(X 5.0) – P(X 3.0). To find the probabilities P(X 3.0) and P(X 5.0) using JMP, we proceed as follows: (Note that JMP allows calculating the exponential probabilities by using the gamma distribution with shape parameter equal to one.)

1. From the main JMP taskbar, select File > New > Data Table, which results in an untitled data table in a separate screen. To assign a title to this table, put the cursor in the box titled “Untitled” in the left corner, right click twice, and enter the desired title.

2. Create two columns, Column 1 and Column 2, in a New Data Table.

3. Enter the values 3 and 5 in Column 1.

4. Recall that Hence label Column 2 as Gamma (1, 0.5) and right-click on Gamma (1, 0.5). Then select the Formula option. Another dialog box Gamma (1, 0.5) appears.

5. In the dialog box Gamma (1, 0.5) select the Probability option from the Function (grouped). Scroll down the drop down menu and select the Gamma Distribution option.

6. In the text box below Gamma Distribution [x] shows up in the Text box. Select Column 1 from the Table Columns, which appears within the parenthesis. After selecting Column 1, enter a comma, and then enter shape parameter = 1. After designating the shape parameter, hit the ‘enter’ key, enter a comma, and then enter the scale parameter = 0.5.

7.Select OK to have the Gamma probabilities calculated in Column 2.

8. In the Table window, the values for P(X 3.0) and P(X 5.0) show

up in the column labeled Gamma (1,0.5) in the JMP printout. Thus, P() = 1P(X 3.0) = 199752125 = 0.00247875, P(X 5.0) = 0.9999546, and P(3.0 X 5.0) = P(X 5.0) – P(X 3.0) = 0.9999546 – 0.99752125 = 0.00243335.

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Example 5.11.3 (Gamma Probabilities Using JMP) The lead time (in days) for orders by Company A of a certain part from a manufacturer is modeled by a gamma distribution with shape parameter= 9 and scale parameter = 4. Find the probability that the lead time for an order is less than or equal to 45 days.

Solution: In order to determine the probability P(X 45) using JMP, we proceed as follows:

1.  From the main JMP taskbar, select File > New > Data Table, which results in an untitled data table in a separate screen. To assign a title to this table, put the cursor in the box titled “Untitled” in the left corner, right click twice, and enter the desired title.

2.  Create two columns, Column 1 and Column 2, in a New Data Table.

3.  Enter the value 45 in Column 1.

4.  Label Column 2 as Gamma (9, 4) and right-click on Gamma (9, 4). Then select the Formula option. Another dialog box Gamma (9, 4) appears.

5.  In the dialog box Gamma (9, 4) select the Probability option from the Function (grouped). Scroll down the drop-down menu and select the Gamma Distribution option.

6.  In the text box below Gamma Distribution, [x] shows up in the Text box. Select Column 1 from the Table Columns, which appears within the parentheses. After selecting Column 1, enter a comma, and then enter the shape parameter = 9. Next, hit the enter key again, followed by a comma, and then enter the scale parameter = 4.

7.  Select OK to have the probability values in Column 2.

8.  In the Table window, the values for P(X 45) show up in the column labeled Gamma (9, 4) in the JMP printout below. Thus, P(X 45) = 0.78945906.

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Example 5.11.4 (Weibull Probabilities Using JMP) From a data set on a system, the parameters of a Weibull distribution are estimated to be a = 4 and b = 0.5 where the time X is measured in thousands of hours. Find the probability P(X £ 10).

Solution: In order to determine the probability P(X 10) using JMP, we proceed as follows:

1.  1. From the main JMP taskbar, select File > New > Data Table, which results in an untitled data table in a separate screen. To assign a title to this table, put the cursor in the box titled “Untitled” in the left corner, right click twice, and enter “Weibull Distribution”.

2. Create two columns, Column 1 and Column 2, in a New Data Table.

3. Enter the value 10 in row 1 of Column 1.

4. Label Column 2 as Weibull (4, 0.5) and right-click on Weibull (4, 0.5). Then

select the Formula option. Another dialog box Weibull (4, 0.5) appears.

5. In the dialog box Weibull (4, 0.5), select the Probability option from the Function

(grouped). Scroll down the drop-down menu and select the Weibull

Distribution option.

6. In the text box below Weibull Distribution [x] will show up in the Text box. Select

Column 1 from the Table Columns, which will appear within the parenthesis. Then

enter a comma, and enter shape parameter = 4. Next, hit the ‘enter’ key again

followed by a comma, and then enter the scale parameter = 0.5.

7. Select OK to have the probabilities calculated in Column 2.

8. In the Table window, the values for P(X 45) show up in the column labeled

Weibull (4, 0.5), in the JMP printout. Thus, P(X 10) = 0.79425934.