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Fill in the empty cells and the values required in the last row of the table. These values will help you answer subsequent questions and calculate the Pearson’s r and the linear regression equation. (2 decimals)

X	Y					()(
3	3	-2.00	-4.00	4.00	16.00	8.00
6	9	1.00	2.00	1.00	4.00	2.00
5	8	0.00	1.00	0.00	1.00	0.00
4	3	-1.00	-4.00	1.00	16.00	4.00
7	10	2.00	3.00	4.00	9.00	6.00
5	9	0.00	2.00	0.00	4.00	0.00
= 5.00	7.00	---------------	---------------	SSx = 10.00	SSY = 50.00	Sum = 20.00

Note: Sum in the last column is .

B. What is the covariance between musical ability and verbal ability? (2 decimals)

COV_XY =

C. Page 4 of the Week 12_2 provides a formula for Pearson’s correlation coefficient that allows you to work with sum of squares (SS_x and SS_Y) instead of having to calculate standard deviations s_X and s_Y . Based on the formula, what is the correlation between musical ability and verbal ability? (2 decimals)

r =

D. Use formulas on p.12 of the Week 12_3 slides to answer the following questions.

a. What is the variance of musical ability? (2 decimals; you may write the final value directly)

=

b. What is the slope (b) for the linear equation line? Use p. 6 of the Week 12_3 slides to write one sentence to interpret the slope. Use the variables names (musical ability and verbal ability) instead of X and Y.

b =

Interpretation:

c. What is the intercept (a) for the linear equation line?

a =

d. Write the linear regression equation.

= bX + a =

Answer 1

A.

x(Musical ability)	y(Verbal ability)	X=x-xbar	Y=y-ybar	(X)^2	(Y)^2	(X)(Y)
3	3	-2	-4	4	16	8
6	9	1	2	1	4	2
5	8	0	1	0	1	0
4	3	-1	-4	1	16	4
7	10	2	3	4	9	6
5	9	0	2	0	4	0
Xbar =5	Ybar =7	S(X)= 0	S(Y)=0	SS(X)= 10	SS(Y)=50	S(XY)=20

Here, n = 6, Xbar =5, Ybar =7, S(X)= 0, S(Y)=0, SS(X)= 10, SS(Y)=50, S(XY)=20, where, X=x-xbar=x-5 and Y=y-ybar= Y=y-7

B. Now, the covariance between Musical ability and Verbal ability

                       = cov(x,y) = Sum[(x-xbar)( y-ybar)]/n = S(XY)/n = 20/6 = 3.33

C. Pearson’s correlation coefficient between Musical ability and Verbal ability

= r(x,y)= Sum[(x-xbar)( y-ybar)]/SQRT[Sum(x-xbar)^2* Sum(y-ybar)^2]

= S(XY)/ SQRT[SS(X)* SS(X)] = 20/ SQRT[10* 50]= 20/22.3606 = 0.89

D. a. Variance of musical ability = Sum(x-xbar)^2/n= SS(X)/n = 10/6 =1.67

b. Slope for the linear equation between Musical ability and Verbal ability is

     b = regression coefficient of Verbal ability on Musical ability

       = Sum[(x-xbar)( y-ybar)]/ Sum(x-xbar)^2 = S(XY)/ss(X) = 20/10 = ½ = 0.5

      Interpretation: Since, the slope b is positive then, Verbal ability is positively elated with

      Musical ability

c. Intercept of linear equation is a = ybar – b*xbar = 7-0.5*5 = 7-2.5 =4.5

d. The linear regression equation is y = bx+a

y = 0.5x+4.5

verbal ability = 0.5 times the musical ability + 4.5