Question

Dr. X was interested in the relationship between optimism and depression. On her depression inventory, high scores correspond to higher levels of depressive symptomology. For optimism, high scores correspond to higher levels of optimism. Her data is presented below.

Optimism Depression

45 13

51 12

35 16

30 17

61 12

54 8

a. Compute the statistic for quantifying the association.

b. Discuss the statistic - what does it tell us about the data.

c. Compute the raw score prediction statistics. Include the linear prediction model (the regression equation.)

d. Compute SST and R^2 (use SST and SSE for R^2).

e. Explain R^2 - what does this quantity tell us?

f. If it were possible to have a score of 0 for optimism, what would the depression score be?

g. Describe how much the depression scores change for each change of 1 on the optimism measure.

h. Use what you computed in c to compute the predicted depression score for a person who scored 40 on the optimism measure.

i. Construct a graph that includes the intercept, the means, and the regression line.

j. Explain the relationship between these two variables to a person who has never had a course in Statistics.

Answer 1

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.827646596
R Square	0.684998888
Adjusted R Square	0.60624861
Standard Error	2.023614205
Observations	6
ANOVA
	df	SS	MS	F	Significance F
Regression	1	35.6199422	35.6199422	8.698367887	0.041998605
Residual	4	16.3800578	4.095014451
Total	5	52
	Coefficients	Standard Error	t Stat	P-value	Lower 95%
Intercept	23.43641618	3.633765512	6.449622604	0.002974612	13.34746571
optimism	-0.226878613	0.07692627	-2.94929956	0.041998605	-0.440460177

a)

r = -0.8276

b)

this means there is strong negative correlation

c)

depression^ = 23.4364 -0.2269 * optimism

d)

SST = 52

R^2 = 35.62/52=0.684998

e)

this means 68.5 % of variation is explained by this model

f)

x = 0

y^ = intercept = 23.4364

g)

when optimism increase by 1 unit, depression decrease by 0.2269 units

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