Question

A study was done to look at the relationship between the number of lovers college students have had in their lifetimes and their GPAs. The results of the survey are shown below.

Lovers	0	2	6	6	1	5	2	0	5
GPA	3.4	3.6	2.9	2.2	3.7	3.1	3.6	3.2	2.8

x value   y value

1. Find the correlation coefficient: r=r=    Round to 2 decimal places.
2. The null and alternative hypotheses for correlation are:
   H0:H0: ? ρ r μ  == 0
   H1:H1: ? ρ μ r   ≠≠ 0
   The p-value is:    (Round to four decimal places)
   
3. Use a level of significance of α=0.05α=0.05 to state the conclusion of the hypothesis test in the context of the study.
   Select an answer At the 5% significance level, the data provides insignificant evidence to conclude that there is a correlation between the number of lovers students have had in their lifetimes and their GPA. Thus, the use of the regression line is not appropriate. At the 5% significance level, the data provides significant evidence to conclude that a student who has had more lovers will have a lower GPA than a student who has had fewer lovers. At the 5% significance level, the data provides insignificant evidence to conclude that a student who has had more lovers will have a lower GPA than a student who has had fewer lovers. At the 5% significance level, the data provides significant evidence to conclude that there is a correlation between the number of lovers students have had in their lifetimes and their GPA. Thus, the regression line is useful.
4. r2r2 =  (Round to two decimal places)
5. Interpret r2r2 : Select an answer Approximately 57% of the variation of students' GPAs can be explained by the variation in the number of lovers using the best-fit regression line. Approximately 57% of variation in the student's number of lovers is explained by the student's GPA using the best-fit regression line.
6. The equation of the linear regression line is:   
   ˆyy^ =  + xx   (Please show your answers to two decimal places)   
   
7. Use the model to predict the GPA of a college student who as had 8 lovers.
   GPA =  (Please round your answer to one decimal place.)   
   
8. Interpret the slope of the regression line in the context of the question: Select an answer For every additional lover students have, their GPA tends to decrease by 0.15. The slope has no practical meaning since a GPA cannot be negative. As x goes up, y goes down.  
   
9. Interpret the y-intercept in the context of the question: Select an answer The y-intercept has no practical meaning for this study. The best prediction for the GPA of a student who has never had a lover is 3.6. The average GPA for all students is predicted to be 3.6. If a student has never had a lover, then that student's GPA will be 3.6.

Answer 1

S.No х y (y-7) 0 2 6 6 1 1 2 3 4 5 6 7 8 9 Total (x-) 9.0000 1.0000 9.0000 9.0000 4.0000 4.0000 1.0000 9.0000 4.0000 50.0000 SSX 3.4 3.6 2.9 2.2 3.7 3.1 3.6 3.2 2.8 28.5 5 2 0 5 (X-X)(y-7) -0.7000 -0.4333 -0.8000 -2.9000 -1.0667 -0.1333 -0.4333 -0.1000 -0.7333 -7.3000 SXY 0.05 0.19 0.07 0.93 0.28 0.00 0.19 0.00 0.13 1.8600 SSY 27 Mean 3.000 3.1667 sample size ni 9 x = 3.0000 T 3.1667 S. (X-X) = Sw (X-7)(y-7)= slope= , S./S.- intercepta B. = V-B,x= 50.000 -7.300 -0.1460 Co 3.6047

correlation r='Sxy/(√Sxx*Syy) =

-0.76

null hypothesis: Ho:           ρ'=		0
Alternate Hypothesis: Ha: ρ≠		0

test stat t=	r*(√(n-2)/(1-r2))=	-3.0649
P value    =	0.0182	(from excel:tdist(3.065,7,2)

At the 5% significance level, the data provides significant evidence to conclude that there is a correlation between the number of lovers students have had in their lifetimes and their GPA. Thus, the regression line is useful.

r² =0.57

Approximately 57% of the variation of students' GPAs can be explained by the variation in the number of lovers using the best-fit regression line.

y^ =3.60+(-0.15)*x

predicted val=3.6+8*-0.15=2.4

For every additional lover students have, their GPA tends to decrease by 0.15.

The best prediction for the GPA of a student who has never had a lover is 3.6.