Question

The following data gives the number of hours 7 students spent studying and their corresponding grades on their exams. Hours Spent Studying 0, 3, 4, 4.5, 5, 5.5, 6 Grades 63, 66, 69, 78, 81, 87, 93. Step 1 of 3: Calculate the correlation coefficient, r. Round your answer to six decimal places.Step 2 of 3: Determine if r is statistically significant at the 0.01 level.Step 3 of 3: Calculate the coefficient of determination, r2. Round your answer to three decimal places.

Answer 1

Solution-:

1 of 3:

We prepare the following table:

	X	Y	X^2	Y^2	X*Y
	0	63	0	3969	0
	3	66	9	4356	198
	4	69	16	4761	276
	4.5	78	20.25	6084	351
	5	81	25	6561	405
	5.5	87	30.25	7569	478.5
	6	93	36	8649	558
Total	28	537	136.5	41949	2266.5

$n=7,\sum X=28,\sum Y=537,\sum X^2=136.5,\sum Y^2=41949$

ΣΧΥ = 2266.5

X 28 ΣΧ 7
and     $\bar{Y}=\frac{\sum Y}{n}=\frac{537}{7}=76.7143$

Σx? Var(X) = - Χ2 136.5 7 42 = 3.5 η

   and      $Var(Y)=\frac{\sum Y^2}{n}-\bar{Y}^2=\frac{41949}{7}-76.7143^2=107.6327$

$SD(X)=\sqrt{Var(X)}=\sqrt{3.5}= 1.8708$    and        $SD(Y)=\sqrt{Var(Y)}=\sqrt{107.6327}=10.3746$

$Cov(X,Y)=\frac{\sum XY}{n}-\bar{X}*\bar{Y}=\frac{2266.5}{7}-4*76.7143=16.9286$

$\therefore Corr(X,Y)=r=\frac{Cov(X,Y)}{SD(X)*SD(Y)}=\frac{16.9286}{1.8708*10.3746}=0.872196$

2 of 3:

Here,$n=7, \alpha=0.01$, $\therefore df=n-2=7-2=5$

(From critical value table of correlation)

$\therefore Critical Value=0.875 >r=0.872196$

So, r is not statistically significant at the 0.01 level.

3 of 3

The coefficient of determination is ,

$r^2=(0.872196)=0.761$

By using R-Software:

> #1 of 3:
> X=c(0,3,4,4.5,5,5.5,6);X
[1] 0.0 3.0 4.0 4.5 5.0 5.5 6.0
> Y=c(63,66,69,78,81,87,93);Y
[1] 63 66 69 78 81 87 93
> r=cor(X,Y);r
[1] 0.8721964
> round(r,6)
[1] 0.872196
> # Comment:There is highly positive correlation.
> #3 of 2
> #Here, Critical Value=0.875 >r=0.872196 at df=5 and alpha=0.01
> #So, r is not statistically significant at the 0.01 level.
> #3 of 3 Coefficient of determination=r^2
> rr=r^2;rr # coefficient of determination
[1] 0.7607265
> round(rr,3)
[1] 0.761
> #This indicate that there is relation between X and Y.

R-Code:

#1 of 3:
X=c(0,3,4,4.5,5,5.5,6);X
Y=c(63,66,69,78,81,87,93);Y
r=cor(X,Y);r
round(r,6)
# Comment:There is highly positive correlation.
#3 of 2
#Here, Critical Value=0.875 >r=0.872196 at df=5 and alpha=0.01
#So, r is not statistically significant at the 0.01 level.
#3 of 3 Coefficient of determination=r^2
rr=r^2;rr # coefficient of determination
round(rr,3)
#This indicate that there is relation between X and Y.