Question

To estimate the quality and price of bars in the D.C. area, 1,700 bars are randomly selected and rated on a scale of 1 to 3 for quality (1 being low quality and 3 being high quality) and a scale of 1 to 5 for price (1 being cheap and 5 being expensive). The results were as follows:

		Price
		1	2	3	4	5
Quality	1	76	99	104	69	120
	2	172	131	111	205	56
	3	117	70	220	106	44

1. Build a joint probability table based on this data set.
2. What is the probability that a bar selected at random will have a price of greater than or equal to 3 and a quality of 2?
3. What is the expected quality of a randomly selected bar in D.C.?
4. What is the expected price of a randomly selected bar in D.C.?
5. What is the covariance of quality and price for bars in D.C.?
6. What is the correlation between quality and price for bars in D.C.?
7. Does the correlation make sense to you? That is, would you expect the price and quality of bars in DC to have a positive, negligible, or negative relationship? Consider drawing on your own experience.

Answer 1

a. The joint probabilities can be computed by dividing each cell frequency by the grand total G = 1700:

Quality	1	2	3	4	5	Total
1	76	99	104	69	120	468
2	172	131	111	205	56	675
3	117	70	220	106	44	557
Total	365	300	435	380	220	1700

b. From the joint probability table:

	Price
Quality	1	2	3	4	5
1	0.045	0.058	0.061	0.041	0.071
2	0.101	0.077	0.065	0.121	0.033
3	0.069	0.041	0.129	0.062	0.026

Required probability = 0.065 + 0.121 + 0.033

= 0.219

	Price (X)
Quality (Y)	1	2	3	4	5	P(Y)
1	0.045	0.058	0.061	0.041	0.071	0.275
2	0.101	0.077	0.065	0.121	0.033	0.397
3	0.069	0.041	0.129	0.062	0.026	0.328
P(X)	0.215	0.176	0.256	0.224	0.129	1.00

c. Expected quality of a randomly selected bar in D.C

.

= (1)(0.275) + (2)(0.397) + 3(0.328)

= 2.052

d. Expected price of a randomly selected bar in D.C

= (1)(0.215) + (2)(0.176) + (3)(0.256) + (4)(0.224) + (5)(0.129)

= 2.876

e. Covariation between quality and price for bars in D.C.

= (1)(1)(0.045)+(1)(2)(0.058)+....+(3)(4)(0.062)+(3)(5)(0.062) - (2.052)(2.876)

= 5.811 - 5.902

= -0.091

f.

Correlation between quality and price for bars in D.C.

where, Standard deviations of X and Y are computed as follows:

= 1.764

Similarly,

= 0.601

Substituting the values,

= -0.086

g. Here, we find that the correlation although too weak to be significant (close to zero) is negative, which implies that as quality increases, price decreases and vice - versa. However, in practice, we would expect a positive relationship between Price and quality. Hence, we may say that this result, based on the given data hardly makes any sense.