Question

The number of hours 10 students spent studying for a test and their scores on that test is represented in the table below.

Hours spent studying, x	0	2	4	5	5	5	6	7	8	9
Test scores, y	40	51	64	69	73	75	93	90	95	95

1. 3) Describe the type of correlation and interpret the correlation in the context of the data. Be specific in describing the magnitude, direction, and strength of the relationship. (12p)

2. 4) Write out the null and alternative hypotheses and conduct a hypothesis testing for the data. At the 1% level of significance, is your test statistic statistically significant? Briefly explain how you reached your conclusion.
   For this question you must state followings: degrees of freedom, one or two tailed hypothesis testing, the critical value, test statistic, do you reject or fail to reject the null hypothesis. (25p)

3. 5) Find the equation of the regression line for the hours spent studying and test scores of students. What are the intercept (a) and slope (b) values of the regression formula, Ŷ = a+ bX for predicting test scores variable from hours spent studying variable. Write the formula replacing a and b by their respective numerical values (round off each element in the final formula to three decimal places). (15p)

Answer 1

	Hours spent (X)	Test scores (Y)	X * Y	X2	Y2
	0	40	0	0	1600
	2	51	102	4	2601
	4	64	256	16	4096
	5	69	345	25	4761
	5	73	365	25	5329
	5	75	375	25	5625
	6	93	558	36	8649
	7	90	630	49	8100
	8	95	760	64	9025
	9	95	855	81	9025
Total	51	745	4246	325	58811

Part 3)

r = 0.9636

There is positive and strong relationship between hours spent on studying and test scores.

Part 4)

To Test :-
H0 :- ρ = 0
H1 :- ρ ≠ 0

Test Statistic :-
t = (r * √(n - 2) / (√(1 - r2))
t = ( 0.9636 * √(10 - 2) ) / (√(1 - 0.9285) )
t = 10.1927

Test Criteria :-
Reject null hypothesis if t > t(α/2,n-2)

Critical value t(α/2,n-2) = t(0.01/2 , 10 - 2 ) = 3.3554

t > t (α/2, n-2) = 10.1927 > 3.3554

Result :- Reject null hypothesis

There is sufficient evidence to conclude that there is statistically correlation between variables.

Part 5)

Equation of regression line is Ŷ = a + bX
b = ( n Σ(XY) - (ΣX* ΣY) ) / ( n Σ X² - (ΣX)² )
b = ( 10 * 4246 - 51 * 745 ) / ( 10 * 325 - ( 51 )²)
b = 6.880

a =( ΣY - ( b * ΣX ) ) / n
a =( 745 - ( 6.8798 * 51 ) ) / 10
a = 39.413
Equation of regression line becomes Ŷ = 39.413 + 6.880 X