Question

For each of the data sets, create a scatter plot with the regression line and answer these questions. What is r, r², Critical Value (CV)? In general, what does r² mean? What does r² mean specifically for the data set? What is the regression equation? Is there any positive or negative correlation? Interpret b₁. Interpret b₀. Forecast the next year.

Year	Cost (Thousands of dollars)
1	100
2	99
3	102
4	80
5	90
6	82
7	80
8	76
9	85
10	70

Answer 1

X(year)	Y(cost)	(x-x̅)²	(y-ȳ)²	(x-x̅)(y-ȳ)
1	100	20.25	184.96	-61.2
2	99	12.25	158.76	-44.1
3	102	6.25	243.36	-39
4	80	2.25	40.96	9.6
5	90	0.25	12.96	-1.8
6	82	0.25	19.36	-2.2
7	80	2.25	40.96	-9.6
8	76	6.25	108.16	-26
9	85	12.25	1.96	-4.9
10	70	20.25	268.96	-73.8

	ΣX	ΣY	Σ(x-x̅)²	Σ(y-ȳ)²	Σ(x-x̅)(y-ȳ)
total sum	55	864	82.5	1080.4	-253
mean	5.5	86.4	SSxx	SSyy	SSxy

sample size ,   n =   10          
here, x̅ =   5.5       ȳ =   86.4  
                  
SSxx =    Σ(x-x̅)² =    82.5          
SSxy=   Σ(x-x̅)(y-ȳ) =   -253          
                  
slope ,    ß1 = SSxy/SSxx =   -3.066666667          
                  
intercept,   ß0 = y̅-ß1* x̄ =   103.2666667          
                  
so, regression line is   Ŷ =   103.2667   +   -3.0667   *x
                  
SSE=   (Sx*Sy - S²xy)/Sx =    304.53          
                  
std error ,Se =    √(SSE/(n-2)) =    6.1698          
                  
correlation coefficient ,    r = Sxy/√(Sx.Sy) =   -0.8474          
                  
R² =    (Sxy)²/(Sx.Sy) =    0.7181      

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r = correlation = -0.8474

R² = coefficient of determination = 0.7181

in general, R² represents how much percentage of variation in observation of y is explained by variable X

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What does r² mean specifically for the data set here,

Here,71.81 % variation in observations of cost is explained by variable year

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What is the regression equation?

from above calculations

regression eqn is

Ŷ =   103.2667    -3.0667   *x

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Is there any positive or negative correlation

since, the value of r is negative,so correlation is negative

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Interpret b₁. Interpret b₀.

b1--> for 1 unit increase in Year,the cost will get decrease by 3.0667(thousand of dollars)

bo--> when year=0,then cost will be 103.2667(thousand of dollars)

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Forecast the next year

Ŷ =   103.2667    -3.0667   *x

for x=11

Ŷ =   103.2667    -3.0667   *11 = 69.533 (thousand of dollars)