Question

Show work including formulas used and tables, must be done by hand.

1. Using the following table of relation between X & Y (Y is the independent Variable):

xi	-2	2	4	5	4	6	7	8	11
yi	-4	1	3	1	3	8	10	20	30

1. Find the estimated regression equation. Show ALL your calculations and the appropriate formula. WRITE Formula in all the questions below and substitute for the numbers, demonstrate your work.

2. Calculate R². Comment on the Goodness of the Fit. Is it a good fit? Why?

3. Test the hypothesis regarding the significance of the slope β₁. (Use α = 0.05) Show ALL the calculations necessary to get your answer. Hint: You need to calculate t by using all the formulas needed for it using t in the Formula Sheet.

.

4. Calculate the values of F, using formula for F statistics.

5. Draw the scatter diagram and also your estimated regression line in a diagram..

Answer 1

X	Y	XY	X²	Y²
-2	-4	8	4	16
2	1	2	4	1
4	3	12	16	9
5	1	5	25	1
4	3	12	16	9
6	8	48	36	64
7	10	70	49	100
8	20	160	64	400
11	30	330	121	900
Ʃx =	Ʃy =	Ʃxy =	Ʃx² =	Ʃy² =
45	72	647	335	1500

Sample size, n =	9
x̅ = Ʃx/n = 45/9 =	5
y̅ = Ʃy/n = 72/9 =	8
SSxx = Ʃx² - (Ʃx)²/n = 335 - (45)²/9 =	110
SSyy = Ʃy² - (Ʃy)²/n = 1500 - (72)²/9 =	924
SSxy = Ʃxy - (Ʃx)(Ʃy)/n = 647 - (45)(72)/9 =	287

a) Slope, b = SSxy/SSxx = 287/110 = 2.6090909

y-intercept, a = y̅ -b* x̅ = 8 - (2.60909)*5 = -5.0454545

Regression equation :

ŷ = -5.0455 + (2.6091) x

b) Coefficient of determination, r² = (SSxy)²/(SSxx*SSyy) = (287)²/(110*924) = 0.8104

yes it is a good fit.

c) Null and alternative hypothesis:

Ho: β₁ = 0 ; Ha: β₁ ≠ 0

α = 0.05

Slope, b = 2.6091

Sum of Square error, SSE = SSyy -SSxy²/SSxx = 924 - (287)²/110 = 175.190909

Standard error, se = √(SSE/(n-2)) = √(175.19091/(9-2)) = 5.00273

Test statistic:

t = b/(se/√SSxx) = 5.4699

df = n-2 = 7

p-value = T.DIST.2T(ABS(5.4699), 7) = 0.0009

Conclusion:

p-value < α Reject the null hypothesis.

d) SSE = SSyy - SSxy²/SSxx =175.19

SSR = SSxy²/SSxx = 748.8091

F = SSR/(SSE/(n-2)) = 748.8091/(175.1909/7) = 29.9197

e) scatter diagram and estimated regression line in a diagram