Question

The Chamber of Commerce in a Canadian city has conducted an evaluation of 300 restaurants in its metropolitan area. Each restaurant received a rating on a 3-point scale on typical meal price (1 least expensive to 3 most expensive) and quality (1 lowest quality to 3 greatest quality). A crosstabulation of the rating data is shown below. Forty-two of the restaurants received a rating of 1 on quality and 1 on meal price, 39 of the restaurants received a rating of 1 on quality and 2 on meal price, and so on. Forty-eight of the restaurants received the highest rating of 3 on both quality and meal price.

Quality (x)	Meal Price (y)			Total
	1	2	3
1	42	39	3	84
2	33	63	54	150
3	6	12	48	66
Total	81	114	105	300

(a)

Develop a bivariate probability distribution for quality and meal price of a randomly selected restaurant in this Canadian city. Let

x = quality rating

and

y = meal price.

Quality (x)	Meal Price (y)			Total
	1	2	3
1
2
3
Total				1.00

(b)

Compute the expected value and variance for quality rating, x.

expected valuevariance

(c)

Compute the expected value and variance for meal price, y.

expected valuevariance

(d)

The

Var(x + y) = 1.6596.

Compute the covariance of x and y.

What can you say about the relationship between quality and meal price? Is this what you would expect?

Since the covariance is  ---Select--- positive negative zero , we  ---Select--- can can not conclude that as the quality rating increases, so does the meal price.

(e)

Compute the correlation coefficient between quality and meal price. (Round your answer to four decimal places.)

What is the strength of the relationship? Do you suppose it is likely to find a low-cost restaurant in this city that is also high quality? Why or why not?

The relationship between quality and meal price is  ---moderately negative or moderately positive or zero ---- and it  ---is or is not --- likley to find a low cost restaurant in this city that also has high quality.

Answer 1

a)dividing each value with 300 in the table

	y
x	1	2	3	Total
1	0.14	0.13	0.01	0.28
2	0.11	0.21	0.18	0.50
3	0.02	0.04	0.16	0.22
Total	0.27	0.38	0.35	1.00

b)

x	P(x)	xP(x)	x-E(x)	(x-E(x))^2	(x-E(x))^2*p(x)
1	0.280	0.280	-0.940	0.884	0.247
2	0.500	1.000	0.060	0.004	0.002
3	0.220	0.660	1.060	1.124	0.247
total	1.000	1.940			0.496
E(x)	=				1.9400
Var(x)=	E(x^2)-(E(x))^2=				0.4964

c)

y	P(y)	yP(y)	y-E(y)	(y-E(y))^2	(y-E(y))^2*p(y)
1	0.270	0.270	-1.080	1.1664	0.315
2	0.380	0.760	-0.080	0.0064	0.002
3	0.350	1.050	0.920	0.8464	0.296
total	1.000	2.080			0.614
E(y)	=				2.0800
Var(y)=σy=	E(y^2)-(E(y))^2=				0.6136

d)

Cov(x,y) =(Var(x+y)-Var(x)-Var(y))/2 =(1.6596-0.4964-0.6136)/2=0.2748

Since the covariance is positive we   can conclude that as the quality rating increases, so does the meal price.

e)

Correlation coefficient ρ=Cov(X,Y)/√(σx*σy)=

0.4979

The relationship between quality and meal price is    moderately positive and it is not likley to find a low cost restaurant in this city that also has high quality.