Mathematics

Complete the following exercises. Make sure to include ALL hand calculations to receive full credit. Be concise and specific in your

answers. You may choose to type your answers in Times New Roman, 12 point font, double space and use the mathematical equation (Insert

menu) or submit a legible handwritten copy. However, the exam must be submitted electronically via Carmen. Hence, if you do write on the

exam, make sure to submit a pdf copy.

1. If sphericityis violated, what is the bestoption (if any) available to researchers?(2 points)
a. There are no good options, so the researcher should use the results of the usual F test and interpret them with caution.
b. The researcher should use aF test that adjusts the degrees of freedom for the critical F value.
c. The researcher should run a factorial ANOVA with a blocking variable instead of covariate.
d. Perform a trend analysis.

2. In a study of different instructional methods, if the researcher is only interested in comparing the scores of the class taught by

the traditional method againsteach of the classes taught by the alternative methods, which procedure should she use?(2 points)
a. Dunn (Bonferroni) method
b. Tukey HSD
c. Dunnettmethod
d. Trend analysis

3. When analyzing mean differences betweenthree samples, doing all pairwise independentt tests instead of ANOVA using the

level…(2 points)
a. decreases the probability of a Type I error to about 1.67%.
b. increases the probability of a Type I error to about 14%.
c. does not change the probability of a Type I error.
d. the exact Type I error rate cannot be determined from the information provided.

4. For which source of variation (in the ANOVA table below) is the null hypothesis rejected at the .05 level ofsignificance?(2

points)
Source df MS F
A 2 15 3.0
B 3 10 2.0
AB 6 3 0.6
Within 120 5
a. A
b. B
c. AB
d. None of the above.

5. In which of the following situations is the assumption of linearity violated? (2 points)
a. Trend analysis shows that there exists a significant quadratic trend.
b. There is no association between the independent variable and the covariate.
c. The distribution of the covariate is seriously skewed in all groups.
d. The scatterplot of the dependent variable and the covariate approaches an asymptote.
6. Dr.Bellus wants to study how people’s perception of attractiveness is affected by the gender of the subjects, and the facial

expression of the object. Each participant views six headshots (in random order) of the same person, who demonstrates different facial

expressions (joy, surprise, fear, anger, sadness, and disgust) in each picture. The 180 participants are then asked to rate the

attractiveness level of each picture. The rating is the dependent variable. Below is the selected output from the ANOVA. Complete the

ANOVA table. (2 points)

Test of Between-Subjects Effects
Source Type III SS df MS F Critical F Decision
A (between-subjects factor) 226.81 3.0465
S (subject)
Test of Within-Subjects Effects
Source Type III SS df MS F Critical F Decision
B (within-subjects factor) 18802.53 38.068
A*B 0.656
B*S
7. A researcher is interested in understanding the nature of the relationship (e.g., linear, quadratic) between teaching experience

and pedagogical content knowledge measured over time for the same 12 teachers. Perform the appropriate analysis, including all relevant

assumption tests, for the data below. (2 points)
Teacher Experience Time 1 Time 2 Time 3
1 High 16 17 28
2 High 17 23 32
3 High 11 21 30
4 High 15 23 35
5 High 21 32 40
6 High 14 22 31
7 Low 12 19 31
8 Low 17 21 25
9 Low 11 16 24
10 Low 8 14 20
11 Low 14 24 28
12 Low 12 19 25

For Question 8-11 use the Reading 1.sav dataset posted to Carmen. This records data for 6th and 7th graders. Unless otherwise stated, use

α=.05, power = .80, and Type III SS. Where hand calculations are needed, round off figures to 4 decimal places.
8. Examine whether Lunch status and Gender predict 7th grade OAA math scores (Math_gr7), controlling for the students’ 6th grade OAA

scores (Math_gr6). Test relevant assumptions and report and interpret all the results.(2 points)
9. Suppose that the normality assumption did not hold. Perform the nonparametric analog of the analysis conducted in Question 8. Have

the underlying assumptions improved? (2 points)
10. Suppose the homogeneity of slopes was violated in the previous nonparametric ANCOVA. Briefly describe a statistical procedure that

you could utilize to model the effects of Lunch, Gender, and Math_gr6 on the rank of the 7th grade OAA math scores. (2 points)
11. Test the interaction for Question 8, wherein the reduced priced and full priced lunch groups are collapsed into a single group.

That is, simplify the 2 (genders [boys and girls]) x3 (lunch status groups [free, reduced, full]) interaction to a 2×2 interaction (2

genders x 2 lunch groups [free vs (reduced, full)]). Report the adjusted means for Question 8 and use them to complete the table below.

Generate a figure using Microsoft Excel of the 2×2 adjusted means. Paste the figurein this document. Interpret the nature of the

interaction.(2 points)

Lunch status
Gender Free Reduced/Full
Male
Female
For Question 12-20 use the Final.sav dataset posted to Carmen.This dataset consists of survey responses from a student’s parent or

guardian (caregiver). It also includes the student’s pre and post assessment test data for reading and math. Unless otherwise stated, use

α=.05, power = .80, and Type III SS. Where hand calculations are needed, round off figures to 4 decimal places.

12. Determine whether the student’s gender (s_Sex) predicts post-test reading scores (postRead). Examine the homogeneity of variance

and normality assumptions, report the ANOVA table, and interpret the results. Include all relevant SPSS tables and figures to support your

conclusions.(2 points)
13. Predict PostRead based on Computer and Internet using Type I SS. Report the ANOVA table and explain the pattern of significance

and nonsignificance. (2 points)
14. Compute the sample size needed to attain an a priori power of .80for the actual effect size (rounded to 4 decimal places) found

for Internet in Question 13, not the lower bound estimate.(2 points)
15. Examine whether the respondent’s education level (CG1Edu) significantly predicted post reading scores (PostRead). If significance

is found, perform Tukey’s HSD post hoc test and summarizethe pattern of significance. You do not need to include the multiple comparison

table. (2 points)
16. Perform a contrast comparing the results for post reading (PostRead) of caregivers with less than a bachelor’s degree (categories

2-5 on CG1Edu) to those with a bachelor’s or higher degree. (There were no parents for category 1.) Report and interpret the contrast

table.
(2 points)
17. Introduce the baseline reading scores (PreRead) into the ANOVA model examined in Question 15. Report the ANCOVA table and provide

a possible explanation for any differences from the previous results. (2 points)
18. Based on the results for Question 17, compute the lower bound 90% CI for the caregiver’s education level (CG1Edu) and perform at

power analysis on this effect size.(2 points)
19. Determine whether the number of jobs the caregiver currently has (CG1Job) and the amount of federal and state assistance

(Assistance) they receive has a significant impact on student post reading scores (PostRead).Make sure the results can generalizeto the

universe of all possible values for the two predictors. Also note that some combinations of CG1Job and Assistance do not exist. Report and

interpret the results of the ANOVA table.(2 points)
20. Test whether the caregiver’s education (CG1Edu) and gender (CG1Gen) predict student’s post reading (PostRead) controlling for the

number of books owned (Books_Own), including all relevant assumptions. (2 points)
BONUS QUESTION (3 points)
For the following question, use the Reading 1.sav dataset.

Create a blocking variable (BaseMath) for Math_gr6 with 10 categories. (Hint, I would find the values for the 10th percentile, 20th

percentile, all the way to the 90th percentile and use these to define the new variable.) Perform a factorial ANOVA to predict Math_gr7

from Treatment and BaseMath. Report and interpret all results, including the profile plot for the two-way interaction.