Question

The file Hotel Prices contains the prices in British pounds (about US$ 1.52 as of July 2013) of a room at two-star, three-star, and four-star hotels in cities around the world in 2013.

City	Two-Star	Three-Star	Four-Star
Amsterdam	74	88	116
Bangkok	23	35	72
Barcelona	65	90	106
Beijing	35	50	79
Berlin	63	58	76
Boston	102	132	179
Brussels	66	85	98
Cancun	42	85	205
Chicago	66	115	142
Dubai	84	67	111
Dublin	48	66	87
Edinburgh	72	82	104
Frankfurt	70	82	107
Hong Kong	42	87	131
Istanbul	47	77	91
Las Vegas	41	47	85
Lisbon	36	56	74
London	74	90	135
Los Angeles	80	118	200
Madrid	47	66	79
Miami	84	124	202
Montreal	76	113	148
Mumbai	41	72	90
Munich	79	97	115
New York	116	161	206
Nice	69	87	133
Orlando	45	78	120
Paris	76	104	150
Rome	75	82	108
San Francisco	92	137	176
Seattle	95	120	166
Shanghai	22	49	79
Singapore	58	104	150
Tokyo	50	82	150
Toronto	72	92	149
Vancouver	74	105	146
Venice	87	99	131
Washington	85	128	158

e. Compute the covariance between the average price at two-star and three-star hotels, between two-star and four-star hotels, and between three-star and four-star hotels.

f. Compute the coefficient of correlation between the average price at two-star and three-star hotels, between two-star and four-star hotels, and between three-star and four-star hotels.

g. Which do you think is more valuable in expressing the relation-ship between the average price of a room at two-star, three-star, and four-star hotels—the covariance or the coefficient of cor-relation? Explain.

h. Based on (f), what conclusions can you reach about the relationship between the average price of a room at two-star, three-star, and four-star hotels?

Answer 1

Using excel,

e. We may compute covariance for every pair of the three variables using the function "COVAR":

The formula for obtaining covariance between two variables X and Y can be obtained using the formula:

We get,

Cov (Two star, Three star) = 492.28; Cov (Two star, Four star) = 548.15; Cov (Three star, Four star) = 914.17

Covariance between two variables is nothing but the extent to which the two variables are dependent on each other. A positive covariance implies a direct relationship and a higher value would mean a higher level of dependency.It ranges from minus infinity to plus infinity.However, it is affected by the unit of measurement.

Among the three covariances computed above, we find that the dependence between Three star & Four star if far greater compared to the other two pairs.

b. The correlation between pairs of the three variables can be obtained using the function "CORREL":

The formula for obtaining correlation between two variables X and Y can be obtained using the formula:

It is nothing but where SD denote the standard deviations of X and Y.

Correlation is more of a quantitative measure of relation between two variables, expressing not only the direction but also the strength of the relationship. It ranges from -1 to 1; '-' indicating a negative relationship.It is a unit less measure.

We get,

Cor (Two star, Three star) = 0.842; Cor (Two star, Four star) = 0.650; Cor (Three star, Four star) = 0.855

We find that all the three pairs are positively correlated. Also, the figures obtained above suggest that the variables (Two star, Three star) and (Three star, Four star) exhibit a stronger relationship as compared to (Two star, Four star).

g. Comparing the result obtained in e and f, we find that both Covariance and Correlation suggest that all three pairs of the variables suggest a positive dependency / relationship.However, the latter provides a better picture of the relation between the two, since, it is a standardized version of covariance. Also it is independent of both location and scale and hence would depict the relationship with same efficiency between any two variables, whatsoever the unit, unlike the former.

h. Average price at two-star, three-star, and four-star are:

Two star hotel average price

Three star hotel average price

Four star hotel average price

Based on the result obtained in f and the averages above,, we find that the average price increases as the stars increases. And the adjacent stars exhibit a stronger dependency i.e.'Two and Three star' and'Three and Four star' than 'Two and Four star'.