Minitab

ANOVA

One-Way

  1. Enter the data for the different groups into separate columns.

  2. Under Stat, choose ANOVA, then One-way.

  3. In the dropdown select Response data are in a separate column for each factor level.

  4. Click the mouse in the box labeled Responses. Highlight the appropriate columns in the box on the left. Click Select.

  5. Press Options. Change Confidence level if desired.

  6. Press OK and OK.

    One-way ANOVA: C1, C2, C3

    Method
    Null hypothesis  All means are equal
    Alternative hypothesis  Not all means are equal
    Significance level  α = 0.05
    Equal variances were assumed for the analysis.
    Factor Information
    Factor Levels Values
    Factor 3 C1, C2, C3
    Analysis of Variance
    Source DF Adj SS Adj MS F-Value P-Value
    Factor 2 6.500 3.250 0.93 0.430
    Error 9 31.500 3.500
    Total 11 38.000
    Model Summary
    S R-sq R-sq(adj) R-sq(pred)
    1.87083 17.11% 0.00% 0.00%
    Means
    Factor N Mean StDev 95% CI
    C1 4 12.250 1.708 (10.134, 14.366)
    C2 4 14.00 2.16 (11.88, 16.12)
    C3 4 12.750 1.708 (10.634, 14.866)
    Pooled StDev = 1.87083

Two-Way

  1. Enter all the data into C1, one column at a time. Enter the row numbers into C2 and the column numbers into C3. Label the columns as is fitting.

  2. Under Stat, choose ANOVA, then General Linear Model, then Fit General Linear Model….

  3. Use select to input C1 as Responses and C2, C3 as Factors. Select OK.

Binomial Distribution

Binomial Probability Distribution

  1. Set up the worksheet in C1 and C2 as shown below.

  2. Press Calc, Probability Distributions, and Binomial.

  3. Designate Probability. (Alternatively, Cumulative Probability)

  4. Complete the dialog box with Number of trials – "12", Event probability – "0.1", Input column – "x", and Optional storage – "p(x)".

  5. Press OK and read output in C2.

Chi-Square Distribution

Test for Association

  1. Input the data in the Worksheet.

  2. Choose Stat, Tables, and Chi-Square Test for Association.

  3. Select Summarized data in a two-way table from the dropdown.

  4. Under Columns containing the table, input column "C2 Yes" and column "C3 No". Press OK.

    Chi-Square Test for Association: Worksheet rows, Worksheet columns

    Rows: Worksheet rows Columns: Worksheet columns
    Yes No All
    1 208
    288.8
    193
    112.2
    417
    2 387
    300.3
    30
    116.7
    417
    3 476
    481.8
    193
    187.2
    669
    All 1071 416 1487

    Cell Counts
    CellCount
    CellExpected count

    Chi-Square Test
    Chi-Square DF P-Value
    Pearson 170.467 2 0.000
    Likelihood Ratio 187.856 2 0.000

    Note: The first row (labeled ‘Pearson’) under Chi-Square Test in the output corresponds to the methods used in the texts.

Test for Goodness of Fit

  1. Input the data in the Worksheet.

  2. Choose Stat, Tables, and Chi-Square Goodness-of-Fit Test (One Variable)...

  3. Select your Observed column for Observed counts.

  4. Select Proportions specified by historical counts and choose your Expected column as the Input column. Press OK.

    Chi-Square Goodness-of-Fit Test for Observed Counts in Variable: Observed

    Observed and Expected Counts
    Category Observed Historical
    Counts
    Test
    Proportion
    Expected Contribution
    to Chi-Square
    1 10 15 0.142857 15 1.66667
    2 15 15 0.142857 15 0.00000
    3 14 15 0.142857 15 0.06667
    4 16 15 0.142857 15 0.06667
    5 11 15 0.142857 15 1.06667
    6 20 15 0.142857 15 1.66667
    7 19 15 0.142857 15 1.06667
    Chi-Square Test
    N DF Chi-Sq P-Value
    105 6 5.6 0.469

Confidence Intervals

Proportion

Suppose a sample of 410 randomly selected radio listeners revealed that 48 listened to WXQI. Construct the 95% confidence interval.

  1. Choose Stat, select Basic Statistics and then choose 1 Proportion.

  2. Select Summarized data from the dropdown and enter Number of events – "48" and Number of trials – "410".

  3. Click the Options button. Make sure the Confidence level is "95" and select Normal approximation from the second dropdown.

  4. Press OK and press OK again.

t-Interval

  1. Select Stat, then Basic Statistics, and 1-Sample t.

  2. In the dropdown select Summarized data and input the sample size, mean, and standard deviation. (Or if you have the raw data enter into C1 and select One or more samples, each in a column and select C1.)

  3. Select Options and choose your confidence level. For a confidence interval use a ≠ alternative hypothesis.

Two Sample t-Interval

  1. Enter the data for the first population in C1, and for the second in C2.

    Two Sample t-interval Minitab step 1
  2. Go to Stat > Basic Statistics > Paired t.

  3. Choose Each sample is in a column; enter C1 for Sample 1 and C2 for Sample 2. Two Sample t-interval Minitab step 3

  4. Click Options in order to specify the desired Confidence level. Two Sample t-interval Minitab step 4

  5. Click OK on the options window and OK on the main dialog window and the confidence interval is produced in the Session window.

Two Sample Proportions z-Interval

  1. Go to Stat > Basic Statistics > 2 Proportions.

  2. Choose Summarized data and enter x1 for Events for the First sample, and n1 for Trials for the First sample. Then enter x2 for Events for the Second sample and n2 for Trials for the Second sample.

    Two Sample Proportions z-Interval Minitab step 2
  3. Choose Options and enter the desired Confidence level.

    Two Sample Proportions z-Interval Minitab step 3
  4. Click OK on the options and main dialog windows and the results are displayed in the Session window.

Minimum Sample Size

  1. Under the Stat menu, select Power and Sample Size, and then select Sample Size for Estimation

  2. Select the Parameter and then enter an estimate of the specified parameter for Planning Value.

    1. You can use information from a previous study, subject-matter knowledge, design specifications, etc. to determine this Planning Value

  3. In the second dropdown, choose Estimate sample sizes and provide your desired Margins of error for confidence itervals.

  4. Click Options… and input the appropriate confidence level.

  5. Click OK and OK.

Standard Deviation

  1. Under the Stat menu, select Basic Statistics, and then select 1 Variance...

  2. Select Sample Standard Deviation in the dropdown. Then, fill in the boxes labeled Sample size and Sample Standard Deviation.

  3. Click on the button labeled Options... In the pop-up window that appears, specify the confidence level and the ≠ Alternative hypothesis.

  4. Click OK and OK.

Variance

  1. Under the Stat menu, select Basic Statistics, and then select 1 Variance...

  2. Select Sample Variance in the dropdown. Then, fill in the boxes labeled Sample size and Sample Variance.

  3. Click on the button labeled Options... In the pop-up window that appears, specify the confidence level and the ≠ Alternative hypothesis.

  4. Click OK and OK.

Counting

Combination

  1. Go to Calc > Calculator.

  2. Type C1 in the box after "Store result in variable:".

  3. Select Combinations under the All functions drop down box and click Select.

  4. Then input a number to replace "number of items" and a number to replace "number to choose" in the expression. For example, input 15 to replace "number of items" and 13 to replace "number to choose" in order to calculate 15C13.

  5. Click OK. The result will be displayed in row 1 of column C1.

    Minitab combination example

Factorial

  1. Go to Calc > Calculator.

  2. Type C1 in the box after "Store result in variable:".

  3. Select Factorial under the All functions drop down box and click Select.

  4. Then input a number to replace "number of items" in the expression. For example, input 10 to calculate 10!

  5. Click OK. The result will be displayed in row 1 of column C1.

    Minitab factorial example

Permutation

  1. Go to Calc > Calculator.

  2. Type C1 in the box after "Store result in variable:".

  3. Select Permutations under the All functions drop down box and click Select.

  4. Then input a number to replace "number of items" and a number to replace "number to choose" in the expression. For example, input 18 to replace "number of items" and 7 to replace "number to choose" in order to calculate 18P7.

  5. Click OK. The result will be displayed in row 1 of column C1.

    Minitab permutation example

Descriptive Statistics

One Variable

  1. Enter the data into column C1.

  2. Under Stat, choose Basic Statistics, then Display Descriptive Statistics.

  3. In the dialog box, input "C1" under Variables.

  4. Click Statistics to select which statistics to include. Select OK.

  5. Observe the Output Screen for the summary statistics.

Graphs

Bar Charts

  1. Enter the category labels in C1 and the corresponding data counts in C2. The axis labels can be entered in the header column.

  2. Select Graph, Bar Chart.

  3. Select Values from a table from the Bars represent: dropdown and ensure Simple is selected for One column of values.

  4. Select C2 for Graph variables and C1 for Categorical variable.

  5. To add a title, click Labels and enter the title under Title.

  6. Click OK and OK.

Dot Plot

  1. Enter the data into column C1.

  2. Select Graph, Dotplot.

  3. Select One Y, Simple and click OK.

  4. Select C1 for Graph variables.

  5. To add a title, click Labels and enter the title under Title.

  6. Click OK and OK.

Histogram

  1. Enter your data in the first column, C1, of Minitab.

  2. Go to the GRAPH menu and choose Histogram.

  3. Select Simple from the types of histograms and click OK.

  4. When the Histogram – Simple menu appears, select C1 as the Graph variable and click OK to generate the graph.

  5. To change how the classes are displayed you can double-click on one of the numbers on the x-axis, select the Binning tab on the Edit Scale menu, choose Midpoint for the Interval Type, select Midpoint/Cutpoint positions under Interval Definition, and enter the midpoints. To display class boundaries choose Cutpoint for the Interval Type, select Midpoint/Cutpoint positions under Interval Definition and enter the class boundaries. You can also change the title, axis labels, color, and major tick positions.

Line Graph

  1. Enter the category labels in C1 and the corresponding data in C2. The axis labels can be entered in the header column. (Category labels are not required.)

  2. Select Graph, Time Series Plot.

  3. Select Simple and click OK.

  4. Select C2 for Series.

  5. Click Time/Scale and select Stamp for Time Scale. Select C1 for Stamp columns.

  6. To add a title, click Labels and enter the title under Title.

  7. Click OK and OK.

Normal Probability Plot

  1. Input your data in C1.

  2. Select Graph, Probability Plot.

  3. With Single selected press OK.

  4. Input "C1" into Graph variables.

  5. Press OK.

Pareto Chart

  1. Enter the category labels in C1 and the corresponding data counts in C2. The axis labels can be entered in the header column.

  2. Select Graph, Bar Chart.

  3. Select Values from a table from the Bars represent: dropdown and ensure Simple is selected for One column of values.

  4. Select C2 for Graph variables and C1 for Categorical variable.

  5. Click Chart Options and select Decreasing Y under Order Main X Groups By.

  6. To add a title, click Labels and enter the title under Title.

  7. Click OK and OK.

Pie Chart

  1. Enter the category labels in C1 and the corresponding data counts in C2.

  2. Select Graph, Pie Chart

  3. Select Chart values from a table and select C1 for Categorical variables and C2 for Summary variables.

  4. To add a title, click Labels and enter the title under Title.

  5. To display the percentages each slice represents on the graph, click Slice Labels and choose Percent.

  6. Click OK and OK.

Stem-and-Leaf Plot

  1. Enter the data in C1.

  2. Select Graph, Stem-and-Leaf

  3. Select C1 for Graph variables and enter the increment value of the stems.

  4. Click OK and OK.

Hypothesis Testing

z-Test

  1. Enter the sample data into column C1.

  2. Choose Stat, Basic Statistics, and 1-Sample Z. Select Options to set the appropriate Alternative hypothesis and press OK.

  3. Enter "C1" for Variables, and enter the Known standard deviation, if applicable. Check the box for Perform hypothesis test and enter the Hypothesized mean. Press OK.

  4. Observe the session window for the results.

One Proportion z-Test

  1. Select Stat, Basic Statistics, 1 Proportion.

  2. From the dropdown choose Summarized data.

  3. Enter the Number of events and the Number of trials. Check Perform hypothesis test and enter a Hypothesized proportion.

  4. Click Options. Enter a Confidence level, select an Alternative hypothesis, and choose Normal approximation for the Method.

  5. Press OK and OK.

    Test and CI for One Proportions

    Method
    p: event proportion
    Normal approximation method is used for this analysis.
    Descriptive Statistics
    N Event Sample p 90% Lower Bound
    for p
    180 133 0.738889 0.696932
    Test
    Null hypothesis  H0: p = 0.7139
    Alternative hypothesis  H1: p > 0.7139
    Z-Value P-Value
    0.74 0.229

Two Proportion z-Test

Method: Summary Statistics

  1. Select Stat, Basic Statistics, 2 Proportions

  2. From the dropdown choose Summarized data.

  3. Enter the Number of events and the Number of trials for each sample.

  4. Choose Options to adjust the Confidence level, Hypothesized difference, Alternative hypothesis, and Test Method.

  5. Press OK and OK.

    Test and CI for Two Proportions

    Method
    p1: proportion where Sample 1 = Event
    p2: proportion where Sample 2 = Event
    Difference: p1 - p2
    Descriptive Statistics
    Sample N Event Sample p
    Sample 1 72 10 0.138889
    Sample 2 72 8 0.111111
    Estimation for Difference
    Difference 90% Lower
    Bound for
    Difference
    0.0277778 -0.042799
    CI based on normal approximation
    Test
    Null hypothesis  H0: p1 − p2 = 0
    Alternative hypothesis  H1: p1 - p2 < 0
    Method Z-Value P-Value
    Normal approximation 0.50 0.307
    Fisher's exact 0.401
    The test based on the normal approximation
    uses the pooled estimate of the proportion (0.125).

Two Sample t-Test (Independent Samples)

  1. Enter the data for the first sample into C1 and the second sample into C2.

  2. Select Stat, Basic Statistics, 2-Sample t

  3. From the dropdown choose Each sample is in its own column.

  4. Enter "C1" from Sample 1 and "C2" for Sample 2.

  5. Choose Options to adjust the confidence level and alternative hypothesis.

  6. Press OK and OK.

Two Sample t-Test (Dependent Samples, Paired Difference)

Method: Raw Data

  1. Enter the data for the first sample into C1 and the second sample into C2.

  2. Select Stat, Basic Statistics, Paired t

  3. From the dropdown choose Each sample is in a column.

  4. Click the mouse in the box labeled Sample 1. Highlight the appropriate column in the box on the left. Press Select. Click the mouse in the box labeled Sample 2, highlight the appropriate column in the box on the left, and press Select.

    Note that Minitab calculates the paired differences by subtracting the values for the second sample from the values for the first sample, which is the opposite of what we do when we calculate them by hand or using a TI-83/84 Plus calculator.

  5. Choose Options to adjust the Confidence level, Hypothesized difference, and Alternative hypothesis.

  6. Press OK and OK.

    Paired T-Test and CI: Cindy, Roommate

    Descriptive Statistics
    Sample N Mean StDev SE Mean
    Cindy 15 26.67 4.81 1.24
    Roommate 15 27.33 5.69 1.47
    Estimation for Paired Difference
    Mean StDev SE Mean 95% CI
    for μ_difference
    -0.667 2.059 0.532 (-1.807, 0.473)
    µ_difference: mean of (Cindy - Roommate)
    Test
    Null hypothesis H0: μ_difference = 0
    Alternative hypothesis H1: μ_difference ≠ 0
    T-Value P-Value
    -1.25 0.230

Two Sample F-Test

  1. Select Stat, Basic Statistics, 2 Variances.

  2. From the dropdown choose Sample variances.

  3. Enter the Sample size and the Variance for each sample.

  4. Press Options. For the Ratio dropdown choose (sample 1 variance) / (sample 2 variance). Enter a Confidence level, Hypothesized ratio (default is 1), and Alternative hypothesis.

  5. Press OK and OK.

    Test and CI for Two Variances

    Method
    σ12: variance of Sample 1
    σ22: variance of Sample 2
    Ratio: σ1222
    F method was used. This method is accurate for normal data only.
    Descriptive Statistics
    Sample N StDev Variance 90% Lower
    Bound for
    σ2
    Sample 1 20 0.079 0.006 0.004
    Sample 2 23 0.066 0.004 0.003
    Ratio of Variances
    Estimated
    Ratio
    90% Lower
    Bound for
    Ratio
    using F
    1.44186 0.816
    Test
    Null hypothesis H2:  σ12 / σ2 = 1
    Alternative hypothesis H1:  σ12 / σ2 > 1
    Significance level α = 0.1
    Method Test
    Statistic
    DF1 DF2 P-Value
    F 1.44 19 22 0.204

Normal Distribution

Inverse Normal

  1. To find a z- or x-value for a given probability in Minitab, enter the probability in the first column and row.

  2. Go to Calc > Probability Distributions > Normal.

  3. When the Normal Distribution menu appears, select Inverse cumulative probability and enter the Mean and Standard deviation.

  4. Select C1 as the Input column. Click OK, and the probability will appear in the Session window.

Note: The probability is the area under the normal distribution curve to the left of the z-score or x-value calculated by Minitab.

Normal Probability (cdf)

  1. Enter the given x- or z-value in the first column and row.

  2. Go to Calc > Probability Distributions > Normal.

  3. When the Normal Distribution menu appears, make sure Cumulative probability is selected and enter the Mean and Standard deviation.

  4. Select C1 as the Input column. Once you are finished, click OK, and the probability will appear in the Session window.

Note: Minitab only calculates the area under the normal distribution curve to the left of the given z-score or x-value.

Poisson Distribution

Poisson Probability Distribution

  1. Set up the worksheet in C1 and C2 as shown below.

  2. Press Calc, Probability Distributions, and Poisson.

  3. Designate Probability. (Alternatively, Cumulative Probability)

  4. Complete the dialog box with Mean – "5", Input column – "x", and Optional storage – "p(x)".

  5. Press OK and read output in C2.

Regression

Confidence Intervals for Slope and y-Intercept

  1. Enter your X and Y data into two columns, C1 and C2.

  2. Press Stat, Regression, and Regression, then Fit Regression Model.

  3. Enter the Response variable and the Predictor variable (continuous).

    Confidence Intervals for Slope and y-intercept Minitab Step 3
  4. Click Results and then choose Display of results: Expanded tables. Click OK and OK.

    Confidence Intervals for Slope and y-intercept Minitab Step 4
  5. Note in the following example that the confidence intervals for the slope (Constant) and y-intercept (Age) are displayed in the output under the Coefficients heading.

    Confidence Intervals for Slope and y-intercept Minitab Step 5

Linear Regression Fitted Line Plot with Confidence Interval

  1. Enter your data in the worksheet.

  2. Under the Stat menu select Regression, and Fitted Line Plot.

  3. Select the Response (Y) and Predictor (X), make sure the Type of Regression Model is Linear. Click Options and under Display Options check Display confidence interval. Click OK and OK.

Linear Regression Fitted Line Plot with Prediction Interval

  1. Enter your data in the worksheet.

  2. Under the Stat menu select Regression, and Fitted Line Plot.

  3. Select the Response (Y) and Predictor (X), make sure the Type of Regression Model is Linear. Click Options and under Display Options check Display prediction interval. Click OK and OK.

Multiple Regression

  1. Enter the data in columns with the variable names at the top of each column.

  2. Under Stat, choose Regression, then Regression, and Fit Regression Model….

  3. In the dialog box, use Select to input the Reponse variable and the Predictor variables. Select OK.

Regression Prediction Intervals

  1. Enter the data in columns with the variable names at the top of each column.

  2. First run Regression (see Linear Regression or Multiple Regression).

  3. Under Stat, choose Regression, then Regression, and Predict

  4. Enter the individual value(s) you wish to predict for. Use Options… to select the Confidence Level. Select OK and OK.

Simple Linear Regression

  1. Enter your X and Y data into two columns, C1 and C2.

  2. Press Stat, Regression, and Fitted Line Plot. Enter the Response variable and the Predictor variable, and press OK.

  3. Observe the least squares coefficients, R 2 , and standard error, as well as the regression plot.

  4. Press Stat, Regression, and Regression, then Fit Regression Model for additional output. Enter the Response variable and the Predictor variable (continuous), and press OK.

  5. If you would like to make predictions using your linear model, press Stat, Regression, and Regression, then click the Predict option.

  6. In the Predict dialogue box, enter the desired response variable, then enter the column that contains your predictor variable (or enter them manually depending on whatever is easiest).

  7. Click Options at the bottom of the box, and enter the specified confidence level and interval type, and press OK.

  8. Press OK in the main dialogue box to calculate predictions.

Sampling

Random Samples

  1. Press Calc, and select Random Data, and then choose Integer.

  2. Complete the dialog box with Number of rows of data to generate, Store in column(s), Minimum value, and Maximum value.

  3. Observe the random numbers.

t-Distribution

Inverse t

  1. Enter the area to the left of the desired t-value in the first row of column C1. If the area we are given is to the right of the area in between, we must first determine the area to the left.

  2. Go to Calc, Probability Distributions, t.

  3. Select Inverse cumultaive probability and enter the number of degrees of freedom. Select C1 as the input column.

  4. Click OK and the t-value will appear in the Session window.