1.
Tests of Between-Subjects Effects /Dependent Variable:Paper /
Source / Type III Sum of Squares / df / Mean Square / F / Sig. /
Corrected Model / 100467.089a / 2 / 50233.544 / 60.029 / .000 /
Intercept / 194860.922 / 1 / 194860.922 / 232.860 / .000 /
Paper_Avg / 100467.089 / 2 / 50233.544 / 60.029 / .000 /
Error / 101254.901 / 121 / 836.817 /
Total / 416319.750 / 124 /
Corrected Total / 201721.990 / 123
a. R Squared = .498 (Adjusted R Squared = .490)
Paper
Tukey HSD
Paper_Avg / N / Subset
1 / 2 / 3
None / 41 / 11.33
50 lbs / 37 / 30.78
100 lbs / 46 / 77.28
Sig. / 1.000 / 1.000 / 1.000
Means for groups in homogeneous subsets are displayed.
Based on observed means.
The error term is Mean Square(Error) = 836.817.
The data indicate that the form of the question influenced the reported average amount of paper consumed: F (2, 121) = 60.03, MSE = 836.82, p .05. The results of the Tukey Test indicate that the subjects’ estimates of paper use differed in all three conditions. Increasing the average value in the question significantly increased their estimates of the amount of paper consumed.
2.
Within-Subjects Factors /Measure:MEASURE_1 /
Foodtype / Dependent Variable /
1 / Fruits /
2 / Veggies
Tests of Within-Subjects Effects /
Measure:MEASURE_1 /
Source / Type III Sum of Squares / df / Mean Square / F / Sig. /
Foodtype / Sphericity Assumed / .746 / 1 / .746 / .264 / .608 /
Greenhouse-Geisser / .746 / 1.000 / .746 / .264 / .608 /
Huynh-Feldt / .746 / 1.000 / .746 / .264 / .608 /
Lower-bound / .746 / 1.000 / .746 / .264 / .608 /
Error(Foodtype) / Sphericity Assumed / 347.334 / 123 / 2.824 /
Greenhouse-Geisser / 347.334 / 123.000 / 2.824 /
Huynh-Feldt / 347.334 / 123.000 / 2.824 /
Lower-bound / 347.334 / 123.000 / 2.824
Foodtype
Measure:MEASURE_1
Foodtype / Mean / Std. Error / 95% Confidence Interval
Lower Bound / Upper Bound
1 / 2.102 / .196 / 1.714 / 2.491
2 / 2.212 / .116 / 1.982 / 2.442
According to the repeated measures ANOVA above, there is not enough evidence to conclude that Amherst students eat different numbers of serving of vegetables and fruits: F (1, 123) = 0.264, MSE = 2.824, p .05. There is no need to conduct a Tukey test here because we failed to reject the null. But, even if we had rejected the null, a Tukey test would have been unnecessary because there were only two levels of the independent variable. If there are only two levels of an IV, then we know that the significant difference (had there been one) would have been between fruits and vegetables. The conclusion we reached regarding the null is identical to the decision we made in response to question #4 on HW 4.
Paired Samples Test // Paired Differences /
/ Mean / Std. Deviation / Std. Error Mean / 95% Confidence Interval / t / df / Sig. (2-tailed) /
/ Lower / Upper /
Pair 1 / Fruits - Veggies / -.1097 / 2.3765 / .2134 / -.5321 / .3128 / -.514 / 123 / .608
The important thing to notice about the relationship between the Fobs for the ANOVA and tobs for the t-test is that F = t2 (0.264 = (0.514 * 0.514). This relationship occurs because the two tests are almost identical. The only difference is that the measure of variability for the t-test is based on the standard error, which is based on the standard deviation. The denominator of the F-ratio is the MSE, which is based on the SS, which is derived from the variance. The variance is equal to the standard deviation squared, so the F-ratio is equal to the t-value squared. If you understand this, great. If you don’t, just let it go.
3. Bristol-Meyers Squib, the company that makes Pepto-Bismol, wants to capitalize on the Thanksgiving Holiday. Last year, they ran a study to determine whether eating the big Thanksgiving meal at different times of day would influence the experience of indigestion. The data, which represent discomfort ratings at 11:00 pm, are presented in the table below. Higher ratings indicate more discomfort. Conduct a one-way ANOVA to determine if meal time influences indigestion. Be sure to report the results of your F-test in the proscribed manner and to conduct post-hoc tests if warranted. Based on the results of your test, would it be valuable for BMS to try to influence the American people? If so, how?
x / x2 / x / x2 / x / x2
7 / 49 / 4 / 16 / 10 / 100
8 / 64 / 5 / 25 / 8 / 64
7 / 49 / 3 / 9 / 9 / 81
9 / 81 / 5 / 25 / 6 / 36
31 / 243 / 17 / 75 / 33 / 281
Ho: All dinner times equally indigestion inducing
Ha: At least two dinner times differ
Fcrit (alpha = .05, dfn = 2, dfd = 9) = 4.26
SStotal = S(x2) – G2/N
= (243 + 75 + 281) - [(31+17+33)2 / 12]
= 599 - 546.75
= 52.25
SSB = [S(T2/n)] - (G2/N)
= [312/4)] + [S(172/4)] + [332/4)] - 546.75
= 240.25 + 72.25 + 272.25 - 546.75
= 38
SSE = S[S(x2) - (T2/n)]
= (243-240.25) + (75-72.25) + (281-272.25)
= 2.75 + 2.75 + 8.75
= 14.25
Anova Table
Source df SS MS F
Time 2 38.00 19.00 12.00
Error 9 14.25 1.58
Total 11 52.25
Because Fobs < Fcrit, I would reject the null. This means that at least one of the dinner times differs from the others. A Tukey test will be conducted to determine which dinner times are different from one another.
HSD = q Ö(MSE/n)
q (a = .05; k=3, df = 9) = 3.82
= 3.95 Ö(1.583/4)
= 3.95 Ö(.40)
= 3.95 (.63)
= 2.49
Mean 2:00 = 7.75 Mean 4:00 = 4.25 Mean 6:00 = 8.25
Based on the analyses, I would tell Bristol-Meyers that dinner time definitely influences indigestion: F (2, 9) = 12, MSE = 1.58, p < .05. Tukey tests revealed that 4:00 produces less indigestion than either the earlier or later times, which do not differ from one another. Therefore, I would run a commercial campaign that would induce people to eat their big meal either early in the afternoon or in the evening.