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	3	15	48	66
Total	78	117	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	1	2	3	4
2	5	6	7	8
3	9	10	11	12
Total	13	14	15	1.00

(b)

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

expected value 16 variance 17

(c)

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

expected value 18 variance 19

(d)

The

Var(x + y) = 1.6691.

Compute the covariance of x and y.

20

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

Since the covariance is 21---Select---zeronegativepositive , we 22---Select---cancan 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.)

23

Answer 1

a)

dividing each cell value with total 300
	y
x	1	2	3	total
1	0.14	0.13	0.01	0.28
2	0.11	0.21	0.18	0.5
3	0.01	0.05	0.16	0.22
total	0.26	0.39	0.35	1

b)

b)
expected value μ =∑xP(x)=				1.94
variance=∑(x-μ)2P(x)=				0.4964
c)
expected value μ =∑yP(y)=				2.09
variance=∑(y-μ)2P(y)=				0.6019

d)

Cov(x,y) =(Var(X+Y)-Var(x)-Var(y))/2=

0.2854

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

e)

correlation coefficient=Cov/sqrt(Var(x)*Var(y))=

0.5221