Question

The maximum discount value of the Entertainment® card for the "Fine Dining" section for various pages is given in the table below.

Page number	Maximum value ($)
4	16
14	20
25	14
32	17
43	19
57	14
72	16
85	14
90	18

Let page number be the independent variable and maximum value be the dependent variable.

a) Calculate the least-squares line. Put the equation in the form of: ŷ = a + bx. (Round your answers to three decimal places.)

ŷ = (?)  + (?) x

b) Find the correlation coefficient. (Round your answer to four decimal places.)

r =
Is it significant? (Use a significance level of 0.05.)

Yes

No  

c) Find the estimated maximum values for the restaurants on page ten and on page 70. (Round your answers to the nearest cent.)

Page 10:	$ ?
Page 70:	$ ?

d) Does it appear that the restaurants giving the maximum value are placed in the beginning of the "Fine Dining" section? How did you arrive at your answer?

Yes, there is a significant linear correlation so it appears there is a relationship between the page and the amount of the discount.

No, there is not a significant linear correlation so it appears there is no relationship between the page and the amount of the discount.    

e) Suppose that there were 200 pages of restaurants. What do you estimate to be the maximum value for a restaurant listed on page 200? (Round your answer to the nearest cent.) (?)

f) Is the least squares line valid for page 200? Why or why not?

Yes, the line produced a valid response for the maximum value.

No, using the regression equation to predict the maximum value for page 200 is extrapolation.  

g) What is the slope of the least-squares (best-fit) line? (Round your answer to three decimal places.)
(?) Interpret the slope.

As the page number increases by one page, the discount decreases by how much?

Answer 1

X	Y	(x-x̅)²	(y-ȳ)²	(x-x̅)(y-ȳ)
4	16	1839.46	0.20	19.06
14	20	1081.68	12.64	-116.94
25	14	479.12	5.98	53.51
32	17	221.68	0.31	-8.27
43	19	15.12	6.53	-9.94
57	14	102.23	5.98	-24.72
72	16	630.57	0.20	-11.16
85	14	1452.46	5.98	-93.16
90	18	1858.57	2.42	67.06

	ΣX	ΣY	Σ(x-x̅)²	Σ(y-ȳ)²	Σ(x-x̅)(y-ȳ)
total sum	422	148	7680.889	40.222	-124.556
mean	46.889	16.444	SSxx	SSyy	SSxy

a)

sample size ,   n =   9          
here, x̅ =   46.889   ,   ȳ =   16.44  
                  
SSxx =    Σ(x-x̅)² =    7680.889   
SSxy=   Σ(x-x̅)(y-ȳ) =   -124.556  

       
                  
slope ,    ß1 = SSxy/SSxx =   -0.016216          
                  
intercept,   ß0 = y̅-ß1* x̄ =   17.20481          
                  
so, regression line is   Ŷ =   17.205 +   -0.016 *x

b)

correlation coefficient ,    r = Sxy/√(Sx.Sy) =   -0.224

correlation hypothesis test      
Ho:   ρ = 0  
Ha:   ρ ╪ 0  
n=   9  
alpha,α =    0.05

  
correlation , r=   -0.2241  
t-test statistic =    t = r*√(n-2)/√(1-r²) =    -0.6084

p-value =    0.5622  
since, p-value >α=0.05,fail to reject Ho,

hence answer is No, correlation is not significant at α=0.05

  

c)

Ŷ =   17.2048   +   -0.0162   *10 = 17.04

Ŷ =   17.2048   +   -0.0162   *70 = 16.07

Page 10:	$ 17.04
Page 70:	$ 16.07

d)

No, there is not a significant linear correlation so it appears there is no relationship between the page and the amount of the discount.    

e)

Ŷ =   17.2048   +   -0.0162   *x

Ŷ =   17.2048   +   -0.0162   *200=$13.96

f)

No, using the regression equation to predict the maximum value for page 200 is extrapolation.  

g)

least-squares (best-fit) line=-0.016

As the page number increases by one page, the discount decreases by $0.016