Question

"Correlation/Regression QA"

A professor wants to determine the relationship between Middleterm Scores (X) and Final Scores (Y). The professor correlates the MiddleTerm Scores (X) with the Final Scores (Y) for all of his students. The data are presented below.
MiddleTerm Scores:
92, 91, 90, 89, 88, 87, 86, 85, 83, 81, 80, 80
Final Scores:
88, 87, 84, 84, 76, 75, 73, 72, 73, 74, 68, 82

16)What is the relationship between MiddleTerm Scores and Final Scores (Round to two decimal places)

19)What is the value of the observed test statistic (t) used to determine whether this relationship is significantly different from zero (Round to two decimal places)?  

20) Assuming that you want to minimize the probability of a Type I Error of the test and are interested in whether the relationship is positive or negative, what is the critical value used to compare the observed test statistic (t) in order to determine whether this relationship is significantly different from zero?

Answer 1

Using Excel, go to Data, select Data Anlysis, choose Regression.

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.701
R Square	0.491
Adjusted R Square	0.441
Standard Error	3.173
Observations	12
ANOVA
	df	SS	MS	F	Significance F
Regression	1	97.291	97.291	9.661	0.011
Residual	10	100.709	10.071
Total	11	198
	Coefficients	Standard Error	t Stat	P-value
Intercept	51.029	11.289	4.520	0.001
Middleterm Scores	0.448	0.144	3.108	0.011

16. Coefficient of middle term scores = 0.448

With ome unit increase in middle term scores, final scores increase by 0.45.

19. Test statistic = 4.52

20. H0: β = 0

H1: β ≠ 0

Level of significance = 0.05

Degrees of freedom: df = n-2 = 12-2 = 10

Critical value (Using Excel function T.INV.2T(probability,df)) = T.INV.2T(0.05,10) = 2.28

Since test statistic is greater than 2.28, we reject the null hypothesis and conclude that β ≠ 0.

Relationship is significantly different from zero and positive (as Coefficient of middle term scores = 0.448).