# PSY 8625 Capella University Week 8 Analysis of Variance Report Discussion

SECTION 3-GRAPHSFigure 1: Quiz 3 Histogram

Statistics

quiz3

N

Valid

Missing

Mean

Median

Std. Deviation

Variance

Skewness

Std. Error of Skewness

Kurtosis

Std. Error of Kurtosis

Minimum

Maximum

105

0

7.13

7.00

1.600

2.559

-.078

.236

.149

.467

2

10

Figure 2- Quiz 3 Descriptive statistics

Tests of Normality

Kolmogorov-Smirnova

Shapiro-Wilk

Statistic

df

Sig.

Statistic

df

Sig.

quiz3

.143

105

.000

.948

105

.000

a. Lilliefors Significance Correction

Figure 3- Quiz 3 Shapiro-Wilk SPSS Output

Test of Homogeneity of Variances

Levene

Statistic

quiz3 Based on Mean

2.898

Based on Median

2.757

Based on Median

2.757

and with adjusted df

Based on trimmed

2.945

mean

gpa

Based on Mean

.910

Based on Median

.876

Based on Median

.876

and with adjusted df

Based on trimmed

.879

mean

df1

2

2

2

df2

102

102

92.750

Sig.

.060

.068

.069

2

102

.057

2

2

2

102

102

97.314

.406

.420

.420

2

102

.418

Figure 4 Levene’s Test

Tests of Between-Subjects Effects

Dependent Variable: quiz3

Type III

Sum of

Source

Squares

Corrected Model

46.334a

Intercept

301.989

section * gpa

46.334

df

3

1

3

Mean

Square

15.445

301.989

15.445

F

7.097

138.767

7.097

Sig.

.000

.000

.000

Error

219.799

101

2.176

Total

5609.000

105

Corrected Total

266.133

104

a. R Squared = .174 (Adjusted R Squared = .150)

Figure 5, Homogeneity of Regression Slopes

SECTION 4- GRAPHS

Descriptive Statistics

Dependent Variable: quiz3

Std.

section

Mean

Deviation

1

7.27

1.153

2

6.33

1.611

3

7.94

1.560

Total

7.13

1.600

N

33

39

33

105

Figure 6 Descriptive Statistics

ANOVA

quiz3

gpa

Between Groups

Within Groups

Total

Between Groups

Within Groups

Total

Sum of

Squares

47.042

219.091

266.133

1.055

51.765

52.820

Figure 7 ANOVA Output

Tests of Between-Subjects Effects

Dependent Variable: quiz3

df

2

102

104

2

102

104

Mean

Square

23.521

2.148

.528

.508

F

10.951

Sig.

.000

1.040

.357

Type III Sum

Mean

Source

of Squares

df

Square

a

Corrected Model

46.334

3

15.445

Intercept

301.989

1

301.989

section * gpa

46.334

3

15.445

Error

219.799

101

2.176

Total

5609.000

105

Corrected Total

266.133

104

a. R Squared = .174 (Adjusted R Squared = .150)

F

7.097

138.767

7.097

Sig.

.000

.000

.000

Partial Eta

Squared

.174

.579

.174

Figure 8 One Way Ancova

Pairwise Comparisons

Dependent Variable: quiz3

Mean

Difference

(I) section (J) section

(I-J)

Std. Error

Sig.b

1

2

.941*

.336

.018

3

-.632

.350

.221

*

2

1

-.941

.336

.018

*

3

-1.573

.346

.000

3

1

.632

.350

.221

*

2

1.573

.346

.000

Based on estimated marginal means

*. The mean difference is significant at the .05 level.

b. Adjustment for multiple comparisons: Bonferroni.

Figure 9 Post Hoc Results

95% Confidence Interval

for Differenceb

Lower

Upper

Bound

Bound

.123

1.759

-1.485

.220

-1.759

-.123

-2.415

-.732

-.220

1.485

.732

2.415

Figure 10 Quiz 3 Means Plot

Figure 11- Means Plot for GPA

Overview

Our data set contains many variables related to course-specific performance and

overall GPA. In looking at the grades for the course, the teacher sees that

students in one section performed much better on the third quiz than the students

in the other two sections of the course. The teacher is concerned about why

students taking identical quizzes in each section are performing so differently.

She hypothesizes that one section might comprise students with higher overall

GPAs and therefore are stronger students than those in the other two sections.

To see if removing the variance in quiz 3 that is explained by the GPA of the

students, you will perform a one-way ANCOVA using the variables quiz 3, GPA

and section.

Preparation

Refer to Chapter 12, “The One-Way ANOVA Procedure,” in IBM SPSS Statistics 25

Step by Step for additional information on using SPSS.

• If necessary, review the copy/export output instructions to refresh your

memory on how to perform these tasks. As with previous assessments, your

submission should be in narrative format with supporting statistical output

(table and graphs) integrated into the narrative in the appropriate places (not

all at the end of the document).

You will use the following variables in the data set:

•

Quiz 3.

•

Section.

•

GPA.

Instructions

1. Briefly describe the goal of your analysis.

•

Specify the variables used in this analysis (independent, covariate, and

dependent) and the scale of measurement of each variable.

•

State the sample size (N) and the alpha level you will use (.05 unless

otherwise specified).

•

Explain why the analysis is a good choice based on the nature of your

variables.

2. Articulate a null hypothesis and alternative hypothesis. Specify a research

question for each predictor. Specify the alpha level.

3. Articulate and test the assumptions of ANOVA.

•

Paste the SPSS histogram output for Quiz 3 and discuss your visual

interpretations.

•

Paste SPSS descriptives output showing skewness and kurtosis values

for Quiz 3 and interpret them.

•

Paste the SPSS output for the Shapiro–Wilk test of Quiz 3 and interpret

the results.

•

Report the results of the Levene test and interpret it.

•

Present the results of the test of homogeneity of regression slopes.

•

Explain whether the assumptions of ANOVA are met. If assumptions

are not met, discuss how to ameliorate violations of the assumptions.

4. Paste the SPSS output of the means plot and provide an interpretation.

•

Paste the SPSS ANOVA output and report the results of the F test.

Include the following:

▪

Degrees of freedom.

▪

F values.

▪

p values.

▪

Calculated effect size.

▪

Interpretation of the effect size.

▪

Interpret statistical results against the null hypothesis and state

whether it is accepted or rejected.

Provide a brief summary of your analysis and the conclusions drawn.

o

Discuss whether the removal of the variance in Quiz 3 that was

explained by GPA influences the relationship between Quiz 3 and

section.

o

Analyze the strengths and limitations of the statistical test.

o

Provide any possible alternate explanations for the findings and

potential areas for future exploration.

Additional Requirements

•

•

•

Written communication: Should be free of errors that detract from the overall

message.

APA formatting: References and citations are formatted according to current

APA style guidelines. Refer to Evidence and APA for more information on how

to cite your sources.

Length: 8–10 double-spaced pages, in addition to the title page and

references page.

Hints for Week 8 Assignment

Test assumptions

In previous units you have already learned how to test the assumptions of normality (ShapiroWilk test) and equal variances (Levene’s test). In this assignment you need to test one

additional assumption: the assumption of homogeneity of regression slopes.

•

•

•

•

•

Open Analyze: General Linear Model: Univariate

Put quiz 3 into Dependent Variable

Put section into Fixed Factor

Put gpa into covariate.

Press the button Model

In the popup window select Build terms, move section and gpa into Model. Next, select both

variables and click on the blue arrow under interaction. This will create an interaction term of

section and gpa. Click Continue and OK to run it.

In the output check the row section * quiz 3. If the p value (Sig.) is .05 or less, it means that the

interaction term of section and gpa is significant. The effect of one variable on the dependent

variable is inconsistent across different levels of the third variable. In other words, the

regression slopes are NOT the same and thus the assumption is violated. If the p value is

above .05, it means there is no interaction and thus the assumption is met.

Run ANCOVA in SPSS

SPSS “remembers” what you did. To run ANCOVA, you need to remove the previous selections

and run ANCOVA without building the interaction term.

•

•

•

•

Open Analyze: General Linear Model: Univariate

Put quiz 3 into Dependent Variable

Put section into Fixed Factor

Put gpa into covariate.

•

From Options check Descriptive statistics, estimate of effect size, and homogeneity tests

The partial eta squared is the effect size contributed by the independent variable. For

example, if the partial eta squared of section is .1 (I made up this number), it means that by

controlling the pre-existing difference among students in terms of GPA, 10% of the variance of

quiz 3 score can be explained by class section.

Another way is to obtain the effect size is: Corrected model sum of squares/corrected

total sum of squared. In this case, it should be the same as partial eta squared.