Question

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Answer 1

a) Part B follows

The scatter plot is given below

9 200 г 150 100 50 0 10 CD 20 30 40

b) The following results are calculated below in the table

X	Y	X^2	Y^2	XY
1	39	1	1521	39
5	47	25	2209	235
11	73	121	5329	803
16	100	256	10000	1600
26	150	676	22500	3900
36	200	1296	40000	7200
$\sum=$ 95	$\sum=$ 609	$\sum=$ 2375	$\sum=$ 81559	$\sum=$ 13777

= 95

у 609

22375

81559

ry = 13777

$r = \frac{n\sum xy-\sum x \sum y}{\sqrt{n\sum x^{2}-(\sum x)^{2}} \sqrt{n\sum y^{2}-(\sum y)^{2}}}$

6 13777-95 609 уб * 2375ー(95)2 V/6 * 81559-16099)2

r0,997

c)

$\bar{x}=\sum_{i=1}^{n} \frac{x_{i}}{n}=\frac{1+5+11+16+26+36}{6}=\frac{95}{6}=15.83$

$\bar{y}=\sum_{i=1}^{n} \frac{y_{i}}{n}=\frac{39+47+73+100+150+200}{6}=\frac{609}{6}=101.5$

The regression coefficients for least square regression is obtained as follow

$b=\frac{n\sum xy-\sum x\sum y}{n\sum x^2-(\sum x)^2}$

613777-95 609 6 2375 (95)2

b4.75

$a=\frac{\sum y- b\sum x}{n}$

$a=\frac{609-4.75*95}{6}$

$a=26.3$

The regression equation is
Y = 26.3 + 4.75 X

d) Part D follows

The least square line is given below

9 200 г 150 100 50 0 CD 10 20 30 40

e) R² is calculated with the following formula

$R^{2}=1-\frac{SSE}{TSS}$

Where $TSS=\sum (y_{i}-\bar{y})^2=(39-101.5)^2+...+(200-101.5)^2=19745.5$

and $SSE=\sum (y_{i}-\hat{y})^2$ , $\hat{y}= 26.3 + 4.75 X$ and is obtained in the below table

X	Y	$\hat{y}$	$(y_{i}-\hat{y})^2$
1	39	31.05	63.20
5	47	50.05	9.30
11	73	78.55	30.80
16	100	102.3	5.29
26	150	149.8	0.04
36	200	197.3	7.29
$\sum=$ 95	$\sum=$ 609	$\sum=$​​​​​​​ 609.05	$\sum=$​​​​​​​ 115.93

$\therefore SSE=115.93$

$R^{2}=1-\frac{115.93}{19745.5}=0.994$

99.4% explained

and 0.6% is unexplained.

f) Y = 26.3 + 4.75 * 10

Y = 73.8