Minitab on the Math OWL Computers (Windows NT)

Transcription

1 STAT 100, Spring 2001 Minitab on the Math OWL Computers (Windows NT) (This is an incomplete revision by Mike Boyle of the Spring 1999 Brief Introduction of Benjamin Kedem) Department of Mathematics, UMCP Contents 1 Preliminaries Caveat What is Minitab? Campus Locations That Carry Minitab Starting the Minitab Session Closing Minitab More About Minitab Windows Data Considerations Data Input Saving and Deleting Data Minitab Canned Data Random Data Generation Various Calculations/Manipulations Sort and Copy Simple Mathematical Expressions More on Coin Tossing Statistical Description of Data Basic Statistics Graphical Display of Data

2 5 Advanced Topics Inference for a Normal Mean Correlation Preliminaries 1.1 Caveat This is designed to be a very fast and covenient introduction for students working on the Windows NT computers in the Math OWL. It is a revision of the Brief Introduction of Professor Kedem in The section 2.2 on Saving and Deleting Data, and the Advanced Topics sections 5.1 and 5.2 were not (yet?) revised, and should be skipped on a first run through at the OWL. 1.2 What is Minitab? Minitab is a statistical package developed at the Pennsylvania State University to aid in the teaching of statistics. It is a powerful tool for data analysis, graphical display of data, and statistical inference. The student edition for windows is particularly easy to learn and use. You can learn much more than the present very basic introduction by simply getting your hands dirty applying the many self-explanatory options Minitab offers to various data sets. In a typical Minitab session, you can input data or generate data internally, and then apply Minitab procedures to your data. The output may consist of simple mathematical results (sum, minimum, division of each datum by 3, etc.), basic descriptive statistics (sample mean, sample standard deviation, etc.), graphs (histogram, boxplot, stem-and-leaf, etc.), statistical inference (hypothesis test, confidence interval, etc.), or some data manipulation such as saving a data set, creating new data columns from existing ones, sorting data, deleting data, and so on. The purpose of this handout is a simple, fast, and easy introduction to Minitab appropriate for STAT 100. We shall not go into complex Minitab statistical procedures that belong in advanced courses. 2

3 1.3 Campus Locations That Carry Minitab Minitab is available at the Mathematics Open Workstation Lab (OWL), MATH The Math OWL is open Monday through Friday, 9 to 6. You need WAM and print accounts. A print account is obtained by establishing a Terrapin Express account ($ minimum) in Room 1109, South Campus Dining Hall, and then transferring funds to a print account. WAM accounts can be obtained from the Academic Information Technology Center, 1st floor of the new Computer Science building. 1.4 Starting the Minitab Session To start Minitab at the OWL lab, MATH 0203 (Windows NT) click/drag: Start Programs Minitab13 for Windows Minitab Three windows open up: 1. Untitled Worksheet, displaying the menu options File, Edit, Manip, Calc, Stat, Graph, Editor, Window, Help. 2. Session, a window that will display the statistical output. 3. Data, a worksheet with columns and rows, used to input, browse, and edit data. The first thing which catches the eye is the Data window. You will notice that its columns are labeled C1, C2, C3,... and so on. The intersection of a column with a row in the Data window is called a cell. The Data window consists of numerous such cells. Each column is reserved for a variable such as height, weight, grade, student ID, etc Closing Minitab You exit Minitab by clicking File Exit In case the Data window contains new data, Minitab gives the option of saving 3

4 your data (see the section on saving data). In this case the exit sequence is File Exit Save (yes, no, cancel) 1.6 More About Minitab Windows In addition to the Untitled Worksheet, Data, and Session, there are several more windows: Graph, Info, History, Help. All graphs are displayed on Graph windows, and in any one session there can be many such windows, one for each plot. The Info and History windows give various summaries of your Minitab activity. The Help window gives an on-line help regarding all aspects of Minitab. You can copy and paste Help information using commands in the File and Edit menus. The windows can be manipulated and controlled through Window on the main Minitab menu bar. To rearrange overlapping windows click Window to get the corresponding menu and click the desired option. For example, to expose the Data window when masked by other windows, simply click Window Data. Note that a window can be minimized to its icon by clicking on the button with the hyphen in the upper right-hand corner of the window. You can also enlarge and close a window by clicking the appropriate buttons in its upper right-hand corner. To print any facing window, be it a Data window, Session, Graph, or any other window, click: File Print Window. Remark: When the Minitab windows are frozen and most menus are highlighted, look for a message in the Session window to unlock the windows. This can happen for example when Minitab is stuck with a certain operation. 2 Data Considerations 2.1 Data Input The columns in the Data window are automatically labeled C1, C2, C3,... Each column stores data from a single variable. For example, suppose you have 10 weight observations W 1,..., W 10, 10 heights H 1,.., H 10, and 10 waists W ST 1,..., 4

5 W ST 10, obtained from 10 individuals. You may choose any column to input the 10 weights, any column for the heights, and any column for the waists. Thus you may store weights in C1, heights in C2, and waists in C3. To enter a datum into a cell, click on the cell to activate it and type the datum, a datum per cell. Move the data-entry arrow from cell to cell using the mouse, the tab key (across columns at a fixed row), or better the keyboard arrows to move right, left, up, and down. After entering the data of a certain variable in a column, click on the next empty cell in that column to let Minitab know the column is complete. It is like adding an extra empty entry to your data column. This is necessary because otherwise you will not be able to make use of the data. You may name the columns (i.e. variables) by typing a name at the heading of each column in the name row just below the C s. For example, we can name the weights column W, in which case Minitab will use this name to identify the weight variable. Likewise, we may name the height variable H and the waist Waist. The default names are the C s supplied by Minitab: C1,C2,C3, etc. Renaming the columns does not eliminate the C s. 2.2 Saving and Deleting Data [I have not tried to review this section, and you don t need it to get going in the OWL skip it for now unless you need it for your personal computer. MB] Note the following: 1. Minitab data file names always end with mtw. As a reminder of this, when you are asked to supply a file name in the space called File Name, you encounter *.mtw. Simply replace the * with a data name such as data1. The resulting name is data1.mtw. Another way to enter the name is to completely erase *.mtw and type instead any name such as data1. Minitab will subsequently add the suffix mtw. 2. You can choose between two drives, c: and a:. The default is the c: drive that stores the data in the Minitab folder data. Choosing drive a: allows you to save the data on a 3.5 disk. To save the data you enter in the Data window in the folder called data, click 1 1 A campus lab may not allow you to save on the hard disk for more than a single 5

6 File Save Worksheet Select File Type file name in File Name Suppose the file name is lifexp. Then the file resides on the hard disk along the path c:\mtbsvw\data To delete lifexp using Windows 95, click: Start Run Provide the above path lifexp File Delete Yes This places lifexp in the Recycle Bin. You can now delete lifexp from the Recycle Bin if you wish to. To save your data for future use on a 3.5 disk the sequence is File Save Worksheet Select File Type file name in File Name Choose a: in Drives With data at hand, you are now ready to do data manipulations and statistics. 2.3 Minitab Canned Data You can retrieve canned data that Minitab stores in the folder Data mentioned above. That directory is home to many useful data sets to which you can apply all the statistical functions of Minitab. (You may appreciate the convenience of not having to type in, or load from some floppy, sets of data to practice on.) Each data set is a file whose name ends with mtw. For example acid.mtw, alfalfa.mtw, cholest.mtw, cities.mtw, plywood.mtw, and so on. To load any of these Minitab supplied data sets say Grades.mtw onto the Data window for statistical use, follow the sequence: File Open Worksheet (data file names appear) session. 6

7 Click a data set name; e.g. Grades Open The data from Grades now appear in the Data window in 3 columns called Verbal, Math, SAT (if I remember corrrectly). 2.4 Random Data Generation A useful feature of Minitab is random data generation where the data follow a particular behavior referred to as distribution in statistics. You can generate data from quite a few distributions. The following example is an important special case. Suppose we wish to imitate a sequence of 10 coin tosses where the chance of a head is 50%. Such a sequence is referred to in statistics as a sequence of Bernoulli trials where the chance of success (that is Head ) on any trial is 50%. To get Minitab to generate such a sequence of length 10 we start from Calc in the menu bar: 7

8 Calc Random Data Bernoulli Type 10 in Generate Type C1 in Store in Column(s) Enter 0.5 in Probability of success The binary sequence is now loaded into Column 1. The observed proportion of 1 s you get need not be 50%; Minitab is mimicking the effect of flipping a fair coin, and you do not expect to see 5 heads in every sequence of 10 flips of the coin. For the observed proportion of heads to be close to 50% we need a long sequence of tosses. This will be demonstrated when we speak of data manipulation in the next section. 3 Various Calculations/Manipulations Minitab enables the application of various functions to the data in the Data window to create new variables stored in specified columns, copy columns, erase or modify rows and columns, stack columns on top of each other, and more. These functions appear under Manip and Calc in the menu bar. We already saw that random data generation is available through Calc. In the following we describe a few useful additional calculations/manipulations possibilities as typical examples. These require no knowledge of statistics. 3.1 Sort and Copy Start with a column C1 of numerical data as you wish. Here are some sample manipulations. Suppose we wish to sort (arrange the data from smallest to largest) column C1 in column C2, and also copy C1 to column C3, as in Table??. To sort: Manip Sort Type C1 in Sort column(s) Type C2 in Store sorted column(s) in Type C1 in Sort by column 8

9 To copy: Manip Copy columns Type C1 in Copy from columns Type C3 in To columns 3.2 Simple Mathematical Expressions To produce a new column C4 which in each row is the column3 entry minus the column2, do the following: Calc Calculator Type C4 in Store result in variable: Type the expression C3-C2 in Expression To compute the sum of all the numbers in column C2 divided by 35 plus the maximum number in column C3 and store the result in C5, do: Calc Calculator Type C5 in Store result in variable: Type the expression Sum(C1)/35 + Max(C2) in Expression The answer of appears in C5. You can experiment with other arithmetic manipulations of columns, and arithmetic functions of columns. 9

10 3.3 More on Coin Tossing Recall the random coin tossing data discussed in Section 2.4. It was remarked there that the proportion of heads in a long sequence of coin tossing is close to 50% provided the coin is balanced. We can actually illustrate this statistical regularity by first generating sufficiently long binary sequences representing sequences of heads and tails of a balanced coin, and then computing their heads proportion following the preceding example. Suppose the 0-1 sequence is stored in C1 and its length is 25. To find the proportion of 1 s click: Calc Calculator Type C2 in Variable (new or modified) Type the expression Sum(C1)/25 in Expression. Repeat this with 100 in place of 25; then with It is likely that the proportion of 1 s you see will be closer to.5 as the sequence length grows. 4 Statistical Description of Data 4.1 Basic Statistics Statistical procedures are listed under Stat in the menu bar. Many useful procedures are available and you can apply any of them to your data, real or computer generated. To get the most basic statistics 2 consisting of the sample mean, standard deviation, median, 1st and 3rd quartiles, minimum and maximum observations, standard error of the sample mean, and trimmed mean, for each column (that is for each variable), click on Stat, then Basic Statistics, and then on the application Descriptive Statistics. Highlight the variables (one or more) of interest by clicking on them, and then click on Select to tell Minitab you are interested in the statistics of the selected variables. Then click OK to get the basic descriptive statistics in the Session window. You can then print the window content as described in Section Consult reference [1] regarding the definitions of these statistics. 10

11 Suppose the variables of interest are stored in columns C2,C3,C4. The above can be summarized as follows: Stat Basic Statistics Descriptive Statistics Select (highlight) C2,C3,C4 all at once Select 4.2 Graphical Display of Data Minitab is capable of producing high-resolution plots of frequency histograms, normalized histograms, cumulative frequencies, boxplots, charts, time series, etc. Each Minitab high-resolution graph appears on its own Graph window. Also available are low-resolution plots that are character-based and appear in the Session window. These include character-based histograms, dot plots (diagrams), and stem-and-leaf plots. Before we explain how to get these plots, it is worthwhile to say a few words about some of them. A frequency histogram is a graphical display of a frequency distribution, where each frequency is attached to a class or category. The frequency of a class is the number of observations falling in the class. Each pair consisting of a class and its frequency gives a rectangle whose width is the class size and whose height is the frequency. The collection of all possible such rectangles gives the histogram. It is possible (and usually desirable) to normalize the histogram so that its total area is 1, in which case Minitab refers to it as a density histogram. Histograms are very important, and Minitab gives various routes to histograms. If column C1 contains data, then the sequence Basic Statistics Display descriptive statistics (Type C1 for the variable) Graphs (Select Histogram ) will produce a frequency histogram display for the numbers in column C1. To get a histogram, on a Graph window, for the numbers that appear in column C1 in the Data window, click on Graph then Histogram, then select the variable of interest C1 by first clicking on it then on Select, and finally clicking OK produces the histogram of interest. This sequence uses the default option of Frequency (midpoint) histogram. 11

12 Several histogram options are available in addition to the default frequency histogram. To select the desired one, prior to plotting, click Options. This gives a list of possible options. Click on the desired option. Thus, to get a normalized histogram choose Density by clicking on the corresponding circle, and then click OK. Another OK gives the desired plot. Assume the desired option is, say, a density histogram of C1. The general way of getting optional histogram plots goes like this: Graph Histogram Options Density C1 Select OK A boxplot summarizes the information in the quartiles and the data range. The plot requires 5 numbers: the minimum (smallest observation in the data), 1st, 2nd, and 3rd quartiles, and the maximum (largest observation in the data). For more details concerning boxplots and the definition of quartiles, see reference [1], Sections 2.4 and 2.5. We can get a Boxplot of the data in C1 from the following sequence: Graph Character Graphs Boxplot C1 Select A dot diagram is another form of frequency distribution. It consists of a line with a scale and dots representing the observations. An observation is plotted as a dot above its value on the line. When several dots fall above the same or nearly the same value they are stacked on top of each other. We can obtain a dot diagram by the following sequence, which just replaces Boxplot above with Dotplot: Graph Character Graphs Dotplot C1 Select Replacing Dotplot in this sequence with the appropriate option will give a stem-and-leaf plot, or a (character-based) histogram. 12

13 5 Advanced Topics 5.1 Inference for a Normal Mean In this section we show how to obtain a confidence interval and how to test a simple hypothesis regarding the mean of a normal population, given a random sample from the population. This is an advanced topic discussed in detail in Chapter 8 of Johnson and Bhattacharyya [1]. Reading this section can be postponed until the time when statistical inference is discussed in class. It is assumed here the reader is familiar with the normal distribution and the material in Chapter 8. We shall use a normal random sample of size 25 from N(µ = 5, σ = 10) generated by Minitab. 13

14 To generate n = 25 N(µ = 5, σ = 10) random observations in column C1, follow the clicking/typing sequence: Calc Random Data Normal Type 25 in Generate Type C1 in Store in Column(s) Enter 5 in Mean Enter 10 in Standard Deviation You can now apply to the generated normal data in column C1 any of the Minitab procedures. In particular, to obtain a 95% confidence interval for the true mean µ = 5, follow: Stat Basic Statistics 1-Sample-Z... C1 Select Confidence interval (Click on the circle) Type 95.0 in Level, and 10 in Sigma From our 25 N(µ = 5, σ = 10) observations generated by Minitab, the preceding clicking/typing sequence produced a 95% confidence interval of (2.80, 10.64) as shown in the upper part of Table 1. We see that the 95% confidence interval indeed captures the true mean. Hypothesis tests concerning normal means follow a procedure similar to that of confidence intervals discussed above. Using the same normal data stored in C1, to test the hypothesis H 0 : µ = 5 versus H 1 : µ < 5, the Minitab clicking/typing sequence is: 14

15 Table 1: Confidence interval and the results of a hypothesis test. Stat Basic Statistics 1-Sample-Z... C1 Select Test mean (Click on the circle and enter 5 in the box) Alternative (Choose Not Equal) Sigma (enter 10 in the box) The results, including the P-value of 0.81 are given in the lower part of Table 1. Since the P-value is quite high, we (correctly!) do not reject the null hypothesis. 5.2 Correlation The (sample) correlation coefficient, denoted by r, is a numerical measure that gives the strength of linear relationship between two variables. The value of r ranges between -1 and +1: 1 r 1. Values of r close to +1 or -1 indicate a strong linear relationship. Values of r close to 0 point to a lack of linear relationship. Suppose the variables of interest are stored in columns C1 and C2, respectively. To get the correlation between the two variables, click: Stat Basic Statistics Correlation Select C1 and C2 15

16 Consider for example a sample of bears. It makes sense to suppose that the length and weight of a bear are highly correlated. To see this, we go to the Minitab data set Bears in the folder Data (we get it by clicking Open Worksheet) and load it into the data worksheet. By following the last sequence, we see that r(length, Weight) = 0.875, relatively high as expected. Similarly, r(head.l, Head.W) = References 1 Johnson, R. A. and G. K. Bhattacharyya, Statistics, Principles and Methods, 3rd Ed., Wiley and Sons, New York,