A psychologist is interested in assessing the extent to which caffeine affects 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 (ages 45-55). The researcher selects a random sample of N =middot40 individuals and randomly assigns n = 10 individuals of the four different conditions. The researcher asks the separate groups of individuals to drink one to each cup of caffeinated coffee and then measures their arousal level on The Arousal Scale five minutes after the measures arousal on a scale of 0-20 (low scores indicating lower beverage is consumed. The Arousal Scale 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)
Gender - Levels - Male, Female
Age - Levels - Young Adults, Middle-Aged Adults

b)
Arousal Level

c)
Null hypothesis - H0G: The effect of caffeine on individual's arousal level does not depend on the gender.
H0A: The effect of caffeine on individual's arousal level does not depend on the age.

Alternative hypothesis - H1G: The effect of caffeine on individual's arousal level depends on the gender.
H1A: The effect of caffeine on individual's arousal level depends on the age.

d)
Number of replicates, r = 10
Number of levels for each variable, a = b = 2
CM = (Sum of all observation)^2 / rab = 366^2 / 10*2*2 = 3348.9

Let Ai be the sum of all observations of level i of factor A. These are row sums.
Let Bj be the sum of all observations of level j of factor B. These are column sums.
Let (AB)ij be the sum of all observations of level i of A and level j of B. These are cell sums.

A1 = 140 , A2 = 226
B1 = 261, B2 = 105
(AB)11 = 82, (AB)12 = 58, (AB)21=179 , (AB)22 = 47

SS(A) = (Sum of Ai^2) / rb - CM
= (140^2 + 226^2) / 10*2 - 3348.9 = 184.9

SS(B) = (Sum of Bj^2) / ra - CM
= (261^2 + 105^2) / 10*2 - 3348.9 = 608.4

SS(AB) = (Sum of (AB)ij)^2 / r - CM - SS(A) - SS(B)
= (82^2 + 58^2 + 179^2 + 47^2) /10 - 3348.9 - 184.9 - 608.4
= 291.6

SSE = SS(Total) - SS(A) - SS(B) - SS(AB)
= 1239.1 - 184.9 - 608.4 - 291.6
= 154.2

DF for A (Gender) = a-1 = 2 - 1 = 1
DF for B (Age) = b-1= 2 - 1 = 1
DF for AxB = (a-1)(b-1) = (2-1) * (2-1) = 1
DF for SSE = ab(r-1) = 2*2*(10-1) = 36

MS(A) = SS(A) /DF(A) = 184.9 / 1 = 184.9
MS(B) = SS(B) /DF(B) = 608.4/1 = 608.4
MS(AxB) = SS(AB) / DF(AB) = 291.6 / 1 = 291.6
MSE = SSE/ DF(SSE) = 154.2 / 36 = 4.28

Test statistics
F for A = MS(A) / MSE = 184.9 / 4.28 = 43.2
F for B = MS(B) / MSE = 608.4 / 4.28 = 142.15
F for AxB = MS*AxB) / MSE = 291.6 / 4.28 = 68.13

e)

Critical value of F for df = 1, 36 at significance level = 0.01 is 7.4
Critical value for each effect is 7.4

f)
As the value of F calculated in part (d) is greater than critical value of 7.4, we reject the null hypothesis and conclude that the effect of caffeine on individual's arousal level depends on the gender and age and the interaction between age and gender for the effect on the arousal level is significant.

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