3. A student believes that undergraduates become more sociable across their 4 ye

Question

3. A student believes that undergraduates become more sociable across their 4 years of college. To test this hypothesis, the student recruits a random sample of students from each of the four classes at his college, asking them to complete a measure of sociability. The scores from this measure (by class membership) are shown below (higher scores reflect higher sociability).

First year 2 4 2 5 1 1

Sophomores 4 5 4 5 4 3

Juniors 6 7 5 6 6 7

Seniors 10 9 8 9 8 7

a.Is there a difference between the four classes in terms of sociability? Perform a one-way ANOVA using = .05. Prepare a source table with the appropriate statistics and place in the space below. Show your work.

b.Conduct a Tukey HSD test to determine the specific effects. Show your work.

c.Calculate the effect size (eta squared).

d.Write up the results in sentence form the correct way including the test statistic, (df), p, Tukey

HSD and effect size (See page 314 middle of page) (Don’t need confidence interval)

Explanation / Answer

a. Grand mean of the sociability = 5.33

Total Sum of Squares , SST = (2-5.33)^2 + (4-5.33)^2 + .... + (7-5.33)^2 = 145.3336

Mean of First year = 2.5

Mean of Sophomores = 4.167

Mean of Juniors = 6.167

Mean of Seniors = 8.5

Treatment Sum of Squares (SSTR) = [6*(2.5-5.33)^2 + 6*(4.167-5.33)^2 + 6*(6.167-5.33)^2 + 6*(8.5-5.33)^2]

= 120.6656

Error Sum of Squares (SSE) = SST - SSTR = 145.3336 - 120.6656 = 24.668

Total Mean Squares (MST)  = SST/ N-1 = 145.3336/(24-1) = 6.3188

Mean Square Treatment (MSTR) = SSTR/c-1 = 120.6656/(4-1) = 40.22187

Mean Square Error (MSE)  = SSE/(N-c) = 24.668/(24-4) = 1.2334

F = MSTR/MSE = 40.22187/1.2334 = 32.61

Critical value of F at significance level of 0.05 and numerator df = 3 and denominator df = 20 is 3.098

We reject the null hypothesis as Observed value of F is greater than critical value of F and conclude that the measure of socialbility is different for different classes.

b. HSD for df = 20 and number of treatment is 3.96

HSD for First year and Sophomores = (M1 - M2)/sqrt(MSE/no. of treatments) = (4.167 - 2.5)/sqrt(1.2334/4) = 3.00

HSD for First year and Juniors = (6.167 - 2.5)/sqrt(1.2334/4) = 6.603

HSD for First year and Seniors = (8.5 - 2.5)/sqrt(1.2334/4) = 10.805

HSD for Sophomores and Juniors = (6.167 - 4.167)/sqrt(1.2334/4) = 3.601

HSD for Sophomores and Seniors = (8.5 - 4.167)/sqrt(1.2334/4) = 7.803

HSD for Juniors and Seniors = (8.5 - 6.167)/sqrt(1.2334/4) = 4.20

So, First year and Juniors; First year and Seniors; Sophomores and Seniors and Juniors and Seniors have statistically different sociability measure because there HSD value is greater than the critical value of HSD (3.96)

c) effect size (eta squared) = SSTR/SST = 120.6656/145.3336 = 0.8302

d) df = 3 and 20, p = 6.753843e-08 for F= 32.61, Tukey HSD critical value = 3.96 and effect size = 0.8302

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