A psychologist is interested in assessing the extent to which caffeine affects a
ID: 3296176 • Letter: A
Question
A psychologist is interested in assessing the extent to which caffeine affects an individual's arousal level depending on one's gender as well as one's age. For the "gender" variable, the researcher is interested in comparing Males to Females, whereas for the "age" variable, the researcher is interested in comparing Young Adults (ages 18-35) to Middle-aged Adults (ages45-55). The researcher selects a random sample of N = 40 individuals and randomly assigns n = 10 individuals to each of the four different conditions. The researcher asks the separate groups of individuals to drink one cup of caffeinated coffee and then measures their arousal level on The Arousal Scale five minutes after the beverage is consumed. The Arousal Scale measures arousal on a scale of 0-20 (low scores indicating lower arousal and high scores indicating higher arousal.) Below are the results of the experiment in table format: a) State the two Independent Variables in this research study and their levels. b) State the Dependent Variable in this research study. c) State ALL of the null and alternate hypotheses for this research study (for each effect). d) Calculate ALL of the appropriate test statistics for this research study (for each effect). e) Determine ALL of the critical regions for this test at alpha 0.01 (for each effect). f) What are ALL of the correct decisions with respect to your hypotheses (for each effect)? g) Calculate one measure of effect size (eta^2) for ALL of your F-tests. h) Depending on the result of your interaction hypothesis, conduct the appropriate test of simple main effects in order to determine which level of A depends on which level of B, making sure to note the F critical value (at alpha = 0.01) and obtained F-ratio for each simple main effect (each level). i) Interpret (in words) the results of ALL of your hypothesis tests, including proper statistical notation.Explanation / Answer
a. The independent variables refer to variables that may be related to a variable measured in the the experiment. The independnet variables in question are age and gender. Variable age has two levels young and middle aged and variable gender has two levels male and female.
b. The variable measured in the experiment is called dependent variable. The response or dependent variable in question is individuals's arousal effect.
c. The complet emodel for 2*2 ANOVA is as follows:
E(y)=beta0+beta1x1+beta2x2+beta3x1x2
where, x1=1 if male 0 if not; x2=1 if young 0 if not
Test1. Test for an interaction effect.
H0: beta3=0 (There is no interaction effect)
H1: there is an effect due to interaction.
Test2. Test for an effect due to factor A.
H0:beta1=0 (there is no effect due to gender)
H1:there is an effect due to gender
Test3. Test for an effect due to factor B.
H0:beta2=0 (there is no effect due to age)
H1: there is an effect due to age.
d. Substitute the given information in following formula compute the followings.
SST=(sigma i=1 to 2, sigma j=1 to 2, sigma k=1 to 10 x^2 ijk)-(t...)^2/abn, where, a=levels of factor A, b i slevels of factor B, and n is number of observations for each combinations of factor A and B.
=(12^2+10^2+...+4^2)-(366^2)/(2*2*10)
=4588-366^2/40
=1239.1
SSA=sigma i=1 to 2 ti..^2/bn-(t...)^2/abn, where, ti.. refer to row total
=(140^2+226^2)/(2*10)-366^2/40
=184.9
SSB=sigma j=1 to 2 t.j.^2/an-(t...)^2/abn, where, t.j. refer to column total
=(261^1+105^2)/(2*10)-366^2/40
=608.4
SS(AB)=sigma i=1 to 2, sigma j=1 to 2, tij.^2/n-sigma i=1 to 2 ti..^2/bn-sigma j=1 to 2 t.j.^2/an+(t...)^2/abn, where, tij. refer to cell total
=(82^2+...+47^2)/10-(140^2+226^2)/(2*10)-(261^1+105^2)/(2*10)+366^2/40
=291.6
SSE=SST-SSA-SSB-SS(AB)=154.2
Test statitic for interaction:
F(AB)=MS(AB)/MSE={SS(AB)/(a-1)(b-1)}/{SSE/ab(n-1)}={(291.6/1)/(154.2/36)}=68.077
F(A)=MSA/MSE={SSA/(a-1)}/{SSE/ab(n-1)}={(184.9/1)/(154.2/36)}=43.167
F(B)=MSB/MSE={SSB/(b-1)}/{SSE/ab(n-1)}={(608.4/1)/(154.2/36)}=142.039
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