Generating Random Values

On the main menu bar, click on "Calc". Then place the cursor over "Random Data". Then click on "Exponential...".

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Click in the box next to "Generate".
Type 100

Click in the box under "Store in column(s):"
  (or just press the Tab key).
Type C1

Click in the box next to "Scale:"   (or just press the Tab key).
Type 1000.

Leave the "Threshold:" box at its default value of zero.
[The "Threshold:" parameter is the lowest possible value, which is also the mode for an exponential distribution.]



Click the "OK" button.

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Give column C1 a meaningful name. [Note that your values will not be identical to the values that you see here. Every new generation of random values from a probability distribution will yield a new set of values.] As the sorted version of these data will appear in column C2, give that column the name Sorted.




(Don’t forget to save your work often!)

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Renaming a Data Sheet

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Sorting Data

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Descriptive Statistics

On the main menu bar, click on "Stat". Then place the cursor over "Basic Statistics". Then click on "Display Descriptive Statistics...".

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In the left pane, double click on either variable name. It appears in the top right box. [If you wish to change which items are displayed in the summary statistics, then click on the "Statistics..." button, check the boxes for the desired items and uncheck the other boxes.] Then click on the "Graphs..." button.

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Check the boxes beside "Histogram of data, with normal curve" and "Boxplot of data" Then click on the "OK" button for this window. Click on the other "OK" button.

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These statistics appear in the Session window.



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Two other windows have appeared.
The default boxplot shows four outliers (all more than one and a half interquartile ranges above the upper quartile and some of them more than three interquartile ranges above the upper quartile) and a pronounced positive skew:



Again, by double-clicking on the graph, one can modify various features (not demonstrated here).

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The default histogram incorporates many of our other results:



The random sample illustrated here came from an exponentially distributed population with a true mean of 1,000 hours.   Yet the sample mean is over 1,150 hours.   A good batch, perhaps, but not that unlikely to have occurred by chance, (as we shall be able to prove, after we have studied continuous probability distributions).

The distribution of the sample is clearly inconsistent with a normal distribution (the red bell shaped curve).   This is no surprise, as the true exponential distribution is strongly non-normal.

The positive skew is so strong that, although the sample mean lamp filament lifetime is about 1,150 hours, half of all of the lamp filaments in this sample burned out in less than 730 hours!

Custom Histogram

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We require a true histogram (where the area, not the height, of each bar represents the frequency in that interval), with seven to twelve equal-width bars spanning most of the range (0, 5500).

Bar widths of 500, starting at 0, are ideal.

Rather than starting from scratch, let us modify the default histogram.
Double click on any number by the vertical axis.

On the "Edit Scale" dialog box: Click on the third tab "Type"; Click the third radio button "Density"; and click the "OK" button.

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Back on the histogram, double click on any number by the horizontal axis.

On the "Edit Scale" dialog box: Click on the third tab "Binning". In the upper group, Click the second radio button "Cutpoint". In the lower group, Click the last radio button "Midpoint / Cutpoint positions:". In the text box, enter values, separated by spaces, from 0 in multiples of 500 to the lowest multiple above your highest value. Click the "OK" button.

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If you wish, you may also modify the visual appearance of the bars. Double-click inside any bar. On the first tab "Attributes" of the "Edit Bars" dialog box, click the respective "Custom" radio buttons and amend the bars as you see fit.

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Here is one version of the modified histogram:



You can copy and paste the above results into your Report Pad.
Then save your work and exit.

Open New Project

On the main menu bar, click on "File". Then hover on "New >" and in the new pop-up menu, click on "Project". Minitab will give you a last chance to save the active project. A new project will then open, with no data in the single worksheet.

Simulation Example #2 - Coin Flips

Creation of a Macro File

Some Notes on Macro Syntax

The first two lines of most Minitab global macros are


    gmacro
    macroName

and the last line is


    endmacro

The line indentation and blank lines are not required.
However, as is the case with any type of computer program, a macro is much easier to read if these style conventions are employed.

The comment symbol ("//" in C++ and "'" in Visual Basic) is the hash mark # in a Minitab macro.   All text following # in a line is ignored by Minitab, (even inside an input prompt or output string).   If you want other readers to know how your macro achieves its objectives, then make good use of comments.

To give a name to a column, use the structure

    name cn 'NameOfColumn'

where n is the number of the column in the worksheet in the Data window.

Before naming a column or entering data into it, check if your algorithm requires any existing data to be cleared first.   That will be the case here, if the same macro is called a second time inside the same project.   The command to clear a range of columns is

    erase cm-cn

To output a literal string, use the keyword note.
On a line that starts with note, all text following on that line, (up to the first #, if any), will appear in the Session window exactly as is.

To print the values of constants in the session window, use

    print kn

Columns appear automatically in the spreadsheet in the Data window.

The principle loop structure (essentially a FOR-NEXT loop with unit step size) is

    do k1=k2:k3
        statements
    enddo

Input

This macro needs no input at all from the user.
If you do wish your macro to accept input, then input from the keyboard uses the structure
    set cn;
        file "terminal";
        nobs 1.

The semicolon ";" indicates that the Minitab command is not complete and continues with subcommands on the next line, while the period "." closes the command.

In principle, more than one datum can be input in the same keyboard response, (separated by a space).   In that case the number of data replaces the "1" in "nobs 1.".   However, it is usually safer to ask the user to input the data one item at a time and to use the note method to prompt the user.

Also note how data may be entered directly from the keyboard only into a worksheet column, not into a constant.   Input data may be copied into a constant using the structure
    copy cm kn
where m and n are positive integers.

Time Series Plot

Execution of the Macro File

Start Minitab.

Click anywhere on the Session window to ensure that it is the active window.
On the main menu bar, click on "Editor".
If the MTB> prompt is not present on the right edge of the session window,
then click on "Show Command Line".


This step is necessary in order to invoke macros and to enter data from the keyboard directly into the Session window.

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The Minitab prompt then appears in the far right portion of the Session window.

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In the Command Line pane of the Session window type "%h:/Coins" (or "d:/Coins" or the other drive, path and name you have given to your macro, immediately after the "%" symbol) and press the "Enter" key. [By default, Minitab searches for macros in its own directory on the c: drive.]

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The results now appear.
They will usually be slightly different with each run.
 


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You can copy and paste the above results into your Report Pad.
Then save your work and exit.

Simulation Example #3 - Shared Birthdays

The next example of the simulation of empirical probability is the famous “birthday problem”:   If n randomly chosen people are in a room, then what is the probability that at least two of them will share the same birthday?

Creation of a Macro File

Copy the following block of text and paste it into a blank document in a word processor (even Notepad will suffice).

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gmacro
Birthdays
### Simulation to estimate the probability that at least two of (k1) people
### in a room will share a birthday.   The lowest (k1) for which the exact 
### probability exceeds 1/2 is k1 = 23 .
### Macro created 2003 03 10 and last modified 2003 03 12 by G.H. George

  erase c1-c8           # needed to ensure clear space in the worksheet
### Constants:
### k1 = number of people in the room
### k2 = number of trials
### k3 = trial index number (loop counter)
### k4 = relative frequency of shared birthdays
### Columns:
  name c1 'birthdays'   # birthdays of the (k1) people in this trial
  name c2 'ranks'       # ranks of birthday data in this trial
  name c3 'no ties'     # ranks if no ties (1 to #people)
  name c4 'diff.'       # abs.diff. of ranks with and without ties in this trial
  name c5 'num.shared'  # number of duplicates found in each trial
  name c6 'shared?'     # = 1 if any duplicates found, 0 if none found, for each trial
  name c7 'num.people'  # temporary column for input of k1
  name c8 'num.trials'  # temporary column for input of k2

### Note that keyboard data entry must be into columns, not constants:
  note Enter number of people:
  set c7;
    file "terminal";
    nobs 1.
  copy c7 k1
  note Enter number of trials:
  set c8;
    file "terminal";
    nobs 1.
  copy c8 k2
### Place the first (k1) natural numbers into column c3:
  set c3
    1:k1
    end.

  do k3=1:k2        # conduct the k2 simulations
      random k1 c1;
        integer 1 365.  # assign random birthdays to the k1 people in this trial
      sort c1 c1    # sort the birthdays into ascending order
      rank c1 c2
      let c4=abs(c2-c3) # zero unless shared birthdays are present in this trial
      let c5(k3)=sum(c4)
  enddo

### Each entry of c6 is 1 (true) if the corresponding entry of c5 is positive, 
### 0 (false) otherwise:
  let c6=(c5>0)

  let k4=sum(c6)/k2
  note The estimated probability that at least two people share the same birthday is
  print k4
  note Compare this with the true probability 
  note p = 1 - (365! / {(365-Num.people)!*(365)^(Num.people)})
  note (approximately 0.507 for 23 people)
endmacro

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Save the file as "h:/Birthdays.mac" (or "d:/Birthdays.mac").

If you have Minitab installed on your computer, then save the macro in your Minitab Macros directory, (on some computers, it is at
"C:/Program Files (x86)/Minitab/Minitab 18/English/Macros/"), with the name "Birthdays.mac".

Some Notes on This Macro

Most of the details of this macro are explained by inline comments in the listing above.   One feature deserves additional comment here:   the trick used to detect equal values in a column of data.

Use the "rank" command to create a column showing the ranks of all entries in the column of (birthdays sorted into ascending order).   The n ranks will be exactly the first n natural numbers, unless any ties exist (that is, shared birthdays).   The sum of the absolute differences between the corresponding entries in the ranks column (c2) and the column containing the first n natural numbers (c3) will be zero if there are no ties, but positive if there are one or more ties.   This part of the algorithm is implemented in the lines inside the do loop

      sort c1 c1    # sort the birthdays into ascending order
      rank c1 c2
      let c4=abs(c2-c3) # zero unless shared birthdays are present in this trial
      let c5(k3)=sum(c4)

Each entry in the column c6 represents the result of one simulation.   Its value is 1 if any ties were detected and 0 if not.   By adding up all entries in that column, the total frequency of shared birthdays is calculated.

Execution of the Macro File

Start Minitab.

Click anywhere on the Session window to ensure that it is the active window.
On the main menu bar, click on "Editor".
If the MTB> prompt is not present on the right edge of the session window,
then click on "Show Command Line".


This step is necessary in order to invoke macros and to enter data from the keyboard directly into the Session window.

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The Minitab prompt then appears in the far right portion of the Session window.

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In the Command Line pane of the Session window type "%h:/Birthdays" (or "d:/Birthdays" or the other drive, path and name you have given to your macro, immediately after the "%" symbol) and press the "Enter" key.

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Respond to the first prompt for data.

                                                                                           


Note how the first eight columns of the Worksheet have now been named.

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Respond to the second prompt for data.

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The results now appear.
They will usually be slightly different with each run.



A major advantage of the macro is that you can repeat this long sequence of commands by simply retyping "%d:/Birthdays" at the Minitab prompt and entering values again for the number of people in each room and the number of rooms.

You can copy and paste the above results into your Report Pad.
Then save your work and exit.