Question

Begin by re-writing the problem. Minitab is required; attach or include your output. Type (use Word) your assignment; you may handwrite equations.

1. Semester GPAs are compared for seven randomly chosen students in each class level at Oxnard University. Do the data show a significant difference in mean GPAs?

GPA for Randomly Selected Students in Four Business Majors
Accounting	Finance	Human Resources	Marketing
2.48	3.16	2.93	3.54
2.19	3.01	2.89	3.71
2.62	3.07	3.48	2.94
3.15	2.88	3.33	3.46
3.56	3.33	3.53	3.50
2.53	2.87	2.95	3.25
3.31	2.85	3.58	3.20

1. At the 0.05 level of significance, determine if there is a difference in the mean GPAs by the four business majors. State your hypotheses and show all 7 steps clearly.
2. Give and interpret the p-value.
3. Should Tukey pairwise comparisons be conducted? Why or why not?
4. If appropriate, use Minitab to produce Tukey pairwise comparison. Write a few sentences with your conclusions from those comparisons.
5. Use Levene’s test to determine if the assumption of homogeneity of variances is valid. Give hypotheses, test statistic, p-value, decision and conclusion. Use the 0.05 level of significance.
6. Verify with Minitab by attaching or including relevant output.

Answer 1

SOLUTION

Seven steps Theory:

Step 1: State the Null Hypothesis
the null would be that there will be no difference among GPAs of different subjects (business majors). Specifically in more statistical language the null for an ANOVA is that all the means are the same. We state the Null hypothesis as:
H0: μ1=μ2=...=μk for k levels of an experimental treatment. [Here, k=4]

Step 2: State the Alternative Hypothesis
HA: treatment level means not all equal
A simpler way of thinking about this is that at least one mean is different from all others.

Step 3: Set α
α = probability of Type I Error [Here, α = 0.05]

Step 4: Collect Data

Step 5: Calculate a test statistic
For categorical treatment level means, we use an F statistic, named after R.A. Fisher. The F value we get from the data is labeled F (calculated). [Here, F(calculated) = 3.52]

Step 6: Construct rejection regions
As with all other test statistics, a threshold (critical) value of F is established. This F value can be obtained from statistical tables, and is referred to as F (critical) or F (α,df(treatment), df(error)). [Here, F (α, 3, 24) = 3.00879]

Step 7: Based on steps 5 and 6, draw a conclusion about H0
If the F (calculated) from the data is larger than the F (α), then you are in the Rejection region and you can reject the Null Hypothesis with (1-α) level of confidence.

Note that modern statistical software condenses step 6 and 7 by providing a p-value. The p-value here is the probability of getting an F (calculated) even greater than what you observe. If by chance, the F (calculated) = F (α), then the p-value would exactly equal to α. With larger F (calculated) values, we move further into the rejection region and the p-value becomes less than α. So the decision rule is as follows:
If the p-value obtained from the ANOVA is less than α, then Reject H0 and Accept HA.

MINITAB Output:

Stat Graph Editor Iools Window Help Assistant Basic Statistics Regression ANOVA DOE '11H One-Way Analysis of Means One-Way Determine whether the means of two ol ChartsAB Belanced ANOVA... Quality Tools Reliability/SuvaFully Nested ANOVA... Multivariate Time Series Iables Nonparametrics Interval Plot Equivalence Tests Power and Sample Size Interaction plot General Linear Model or more groups differ General MANOVA... σ2 Test for Equal Variances |-Main Effects Plot

ne-Way AnalysiS of Variance C1 Accounting C2 Finance C3 Human Resources C4 Marketing Response data are in a separate column for each factor leve Responses: Accounting Finance Human Resources' Marketing Graphs... Options... Comparisons... Results Select Storage... Help Cancel OK

Initially we don't know the significance of treatments, so do not tick Tukey's option.

One-Way Analysis of Variance: Comparisons Way A 23 Error rate for comparisons: 5 vel Comparison procedures assuming equal variances Tukey 「Fisher 「Dunnett Control group level: Accounting Г Hsu MCB Best: Largest mean is best 13 Results aphs. Interval plot for differences of means Grouping information -Tests Cancel Help Help Cancel OK

Once we analyze ANOVA and find that atleast one mean is different, then go for Tukey's test by clicking the checkbox.

One-Way Analysis of Variance: Comparisons One-Way 23 Error rate for comparisons: 5 C1 C2 C3 Hum C4 vel Comparison procedures assuming equal variances Tukey 「Fisher Г Dunnett Control group level: Accounting Г HsuMCB Best: Largest mean is best Results aphs. Interval plot for differences of means Grouping information Г-Tests Help Help Cancel OK

MTB > OneWay;
SUBC> Response 'Accounting' 'Finance' 'Human Resources' 'Marketing';
SUBC> IType 0;
SUBC> Tukey 5;
SUBC> GMCI;
SUBC> TGrouping;
SUBC> GIntPlot;
SUBC> GFourpack;
SUBC> TMethod;
SUBC> TFactor;
SUBC> TANOVA;
SUBC> TSummary;
SUBC> TMeans;
SUBC> Nodefault.

One-way ANOVA: Accounting, Finance, Human Resources, Marketing
Method

Null hypothesis All means are equal
Alternative hypothesis At least one mean is different
Significance level α = 0.05

Equal variances were assumed for the analysis.

Factor Information
Factor Levels Values
Factor 4 Accounting, Finance, Human Resources, Marketing

Analysis of Variance
Source DF Adj SS Adj MS F-Value P-Value
Factor 3 1.181 0.3937 3.52 0.030
Error 24 2.687 0.1119
Total 27 3.868

Model Summary
S R-sq R-sq(adj) R-sq(pred)
0.334584 30.54% 21.85% 5.45%

Means
Factor N Mean StDev 95% CI
Accounting 7 2.834 0.505 ( 2.573, 3.095)
Finance 7 3.0243 0.1776 (2.7633, 3.2853)
Human Resources 7 3.241 0.308 ( 2.980, 3.502)
Marketing 7 3.3714 0.2575 (3.1104, 3.6324)
Pooled StDev = 0.334584

As p-value of ANOVA table is obtained as 0.030 which is less than alpha=0.05, we Reject the null hypothesis H0.
Yes, there is a difference in the mean GPAs by the four business majors.

Graphs:

Interval Plot of Accounting, Finance, 95% CI for the Mean 3.75 3.50 3.25 3.00 2.75 2.50 Accounting Finance Human Resources Marketing The pooled standard deviation was used to calculate the intervals.

Residual Plots for Accounting, Finance, Human Resources, Marketing Normal Probability Plot Versus Fits 0.8 90 04 50 04 10 0.0 -0.8 -0.4 3.0 32 34 0.8 Residual Fitted Value Histogram 0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 Residual

Should Tukey pairwise comparisons be conducted? Why or why not?

Yes, Tukey's pairwise comparisons MUST be conducted because we have Rejected the hypothesis of ANOVA carried above. It is essential to know which of the subjects (means) are different.

Tukey Pairwise Comparisons

Grouping Information Using the Tukey Method and 95% Confidence

Factor N Mean Grouping
Marketing 7 3.3714 A
Human Resources 7 3.241 A B
Finance 7 3.0243 A B
Accounting 7 2.834 B

Means that do not share a letter are significantly different.

Here, GPAs for Marketing and Accounting are significantly different.

Interpreting the results
The p-value for the multiple comparisons test is significant only if two or more intervals do not overlap.
The summary plot displays the multiple comparison intervals. If the intervals for two groups do not overlap, then the standard deviations for those groups are significantly different. The Session window displays the Bonferroni simultaneous confidence intervals for the standard deviations of each population. Both the summary plot and the Session window display the p-values for the multiple comparisons test and Levene's test.

Use Levene’s test to determine if the assumption of homogeneity of variances is valid. Give hypotheses, test statistic, p-value, decision and conclusion. Use the 0.05 level of significance.

l Stat Graph Editor Iools Window Help Assistant Basic Statistics Regression ANOVA DOE Control Charts Quality Tools One-Way Analysis of Means.. Balanced ANOVA General Linear Model Fully Nested ANOVA... General MANOVA A: e Rejiability/Survival Multivariate Ac Time Series ge ▶I Test for Equal Variances Tables Nonparametrics | Interval Plot Equivalence TestsMain Effects Plot..Test for Equal Variances Power and Sample SizeInteraction Plo.. or Determine whether the variances or the standard deviations of two or more groups differ.

Test for Equal Variances C1 Accounting C2 Finance C3 Human Resources C4 Marketing Response data are in a separate column for each factor level Responses: ro Accounting Finance Human Resources' Marketing Graphs...1 Options... Select Results Storage... Cancel Help OK

MTB > VarTest 'Accounting' 'Finance' 'Human Resources' 'Marketing';
SUBC> Unstacked;
SUBC> Confidence 95.0;
SUBC> GInterval;
SUBC> NoDefault;
SUBC> TMethod;
SUBC> TBonferroni;
SUBC> TTest.  

Test for Equal Variances: Accounting, Finance, Human Resources, Marketing

Method
Null hypothesis All variances are equal
Alternative hypothesis At least one variance is different
Significance level α = 0.05

95% Bonferroni Confidence Intervals for Standard Deviations
Sample N StDev CI
Accounting 7 0.505267 (0.255039, 1.55632)
Finance 7 0.177563 (0.063906, 0.76705)
Human Resources 7 0.307703 (0.201376, 0.73100)
Marketing 7 0.257451 (0.105066, 0.98082)

Individual confidence level = 98.75%

Tests
Test
Method Statistic P-Value
Multiple comparisons — 0.099
Levene 1.98 0.145

YES, the assumption of homogeneity of variances is valid as Levene's test statistic=1.98 and p-value=0.145 which is greater than alpha, so we accept H0.

Test for Equal Variances: Accounting, Finance, Human Resour, Multiple comparison intervals for the standard deviation, α-0.05 Multiple Comparisons P-Value 0.099 Levene's Test P-Value 0.145 Accounting Finance Human Resources Marketing 0.0 0.2 0.4 0.6 0.8 1.0 1.2 If intervals do not overlap, the corresponding stdevs are significantly different.