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-five of the restaurants received a rating of 1 on quality and 1 on meal price, 42 of the restaurants received a rating of 1 on quality and 2 on meal price, and so on. Thirty 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	45	42	3	90
2	36	57	66	159
3	9	12	30	51
Total	90	111	99	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.

(b)

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

(c)

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

(d)

The

Var(x + y) = 1.51.

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?

(e)

Compute the correlation coefficient between quality and meal price. 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?

Answer 1

a) dividing each value with 300:

	y
x	1	2	3	Total
1	0.15	0.14	0.01	0.30
2	0.12	0.19	0.22	0.53
3	0.03	0.04	0.1	0.17
Total	0.30	0.37	0.33	1.00

a)

marginal distribution of x
x	P(x)	xP(x)	x-E(x)	(x-E(x))^2	(x-E(x))^2*p(x)
1	0.300	0.300	-0.870	0.757	0.227
2	0.530	1.060	0.130	0.017	0.009
3	0.170	0.510	1.130	1.277	0.217
total	1.000	1.870			0.453

expected value =1.87

variance for quality rating =0.4531

c)

marginal distribution of y
y	P(y)	yP(y)	y-E(y)	(y-E(y))^2	(y-E(y))^2*p(y)
1	0.300	0.300	-1.030	1.0609	0.318
2	0.370	0.740	-0.030	0.0009	0.000
3	0.330	0.990	0.970	0.9409	0.310
total	1.000	2.030			0.629

expected value of y =2.03

variance for meal price, y =0.6291

d)

Covar(y,x)=E(XY)-E(X)*E(Y)=

0.2139

relationship between quality and meal price is positive

e)

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

0.4006

strength of the relationship is moderate