An education researcher is interested in students’ performance on a mathematics test for a novel---and generally unfamiliar---concept.  The researcher believes that if students attend a small-group work-session, then they will be more likely to perform better on the mathematics test. 

       This researcher is testing two hypotheses.  The first is that the group work provides extra time on the task and has an effect on students’ mathematics test scores.  The second is that this treatment effect is moderated by the number of languages spoken.  Past research has suggested that there should not be a main effect for being multi-lingual.  However, to control for this possibility, a 2-way (2x2) factorial ANOVA is being conducted to account for any influence from language, the treatment, and the interaction of the two main effects.

       To test the hypotheses, the research has obtained 28 participants:  14 English-only speakers and 14 students who speak English and at least one other language. From each group of 14, 7 students were randomly chosen for the treatment group.  After the treatment, all students were given the same test.  This test has been used in previous research, has a range of 0 to 100, and measures students’ ability to correctly perform the novel mathematics exercise.  (The higher the test score, the more positive the result.)

       Use a 2x2 ANOVA with α=0.01\displaystyle \alpha={0.01} to test the the data and evaluate the hypotheses.  Results are provided below for each group of students.

  Factor B: Treatment
Control:
No Extra Work 		Treatment:
Extra time with Group
Factor A:
Languages 		English only 52.0
52.6
60.9
51.0
52.4
59.1
43.1 		49.2
52.0
55.5
60.5
57.3
52.1
56.1
Multilingual 52.0
54.7
48.1
50.3
50.0
53.1
54.3 		55.4
57.3
59.9
60.5
57.3
75.0
74.0


( 1a )    What is the F value for the treatment effect? 
(Report answer accurate to 2 decimal place.)

( 1b )    With degrees freedom df1=1\displaystyle {d}{f}_{{1}}={1} (for the numerator) and df2=24\displaystyle {d}{f}_{{2}}={24} (for the denominator), what is the p-value for the F value for the treatment effect? 
(Report answer accurate to 4 decimal places.)

( 1c )    Does this support the researcher's hypothesis that the treatment has an effect on ability to solve the mathematics exercise?


( 2a )    What is the F value for the language effect? 
(Report answer accurate to 2 decimal place.)

( 2b )    With degrees freedom df1=1\displaystyle {d}{f}_{{1}}={1} (for the numerator) and df2=24\displaystyle {d}{f}_{{2}}={24} (for the denominator), what is the p-value for the F value for the language effect? 
(Report answer accurate to 4 decimal places.)

( 2c )    Does this support the researcher's assumption that language does NOT have an effect on ability to solve the mathematics exercise?


( 3a )    What is the F value for the interaction effect? 
(Report answer accurate to 2 decimal place.)

( 3b )    With degrees freedom df1=1\displaystyle {d}{f}_{{1}}={1} (for the numerator) and df2=24\displaystyle {d}{f}_{{2}}={24} (for the denominator), what is the p-value for the F value for the interaction effect? 
(Report answer accurate to 4 decimal places.)

( 3c )    Does this support the researcher's hypothesis that the treatment effect is moderated by language?

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