Any “budget” conscious consumer/traveler would want to find a good deal wherever and whenever he or she can, especially when it comes to airfare and rental cars. In tough economic times, bargain hunting is the norm rather the exception. This is an exercise in price comparison for rental cars; focusing on two popular car rental companies Budget and Avis. This is also an opportunity to use the one-sample t-procedures (test and confidence interval) on matched pairs data.
The following table shows the prices found on the websites of Budget Car Rental and Avis Car Rental. The prices are one-day rental prices for full size sedans quoted for the day of December 12, 2011 (a non-holiday Monday) at 35 of the busiest airports in United States. These prices are basic car rental prices without any discount or upgrade.
Matched Pairs Data
The price data in Table 1 are best viewed as matched pairs data. In such data, observations are taken on the same individual (in this case the same airport) under different conditions (one price is for Budget and one price is for Avis). The Budget car rental prices and the Avis car rental prices are said to be dependent samples.
The alternative to thinking of Table 1 as matched pairs data is to view the Budget Prices and the Avis prices as two independent samples and then use two-sample t-procedures to perform the analysis. But this approach is not the best way of using the data. When you are at the Chicago airport, you do not care about the car rental prices at the Tampa airport or any other airport. Just as you only compare prices among the car rental companies in the same airport, we should compare prices within each matched pair. So thinking of the data as matched pairs affords us the best way to compare prices between the two companies in Table 1.
To analyze the data in Table 1, we first take the difference between the Budget rental prices and the Avis rental prices (Avis minus Budget). These 35 differences form a single sample (the last column in Table 1). The first difference is -$3.29 (indicating that Avis is cheaper by this amount at the Atlanta airport). The second difference is $45.91 (indicating that Budget is cheaper by this amount at the Chicago airport). Most of the calculations and analysis will be done using this “differenced” sample. Thus, the comparative design of using matched pairs data makes use of single-sample procedures (in this case, the one-sample t-procedures).
Initial Look of the Data
Most of the differences are positive (i.e. Avis charges more than Budget). Some of the differences are small. But some of the differences are in the $30 to $50 range. So we need to take a closer look.
The following table shows the sample means and sample standard deviations for the Budget prices, Avis prices and the differences. In these 35 airports, the average one-day rental prices for Budget is $60 and the average Avis price is $73.74. The price differential is $13.64, meaning that Avis price is 22.7% over the Budget average price. Any “budget” conscious traveler should care about a difference of $13. The question is: is the price differential we are seeing statistically significant? Specifically, are the data in Table 1 evidences that Budget Car Rental is less expensive than Avis?
The Requirements for Using the t-Procedures
The use of the t-procedures (confidence interval and test) rests on two assumptions. One is that the data set is a simple random sample. The second is that the distribution of the data measurements has no outliers and follows a normal distribution (or that the sample size is large).
The car rental prices in Table 1 are not a random sample. They are just car rental prices from Budget and Avis at the 35 busiest airports in United States. Because these busy airports spread out across various regions in the United States and because they are of varying sizes, we feel that the sample car rental prices indicated here are representative of car renting experiences at these airports. For these reasons, we feel that there is value in carrying this comparison.
Because the sample size is relatively large (), the need for checking normality assumption is not critical. The car rental prices do not seem to have any extreme data values.
To carry out the t-test and t-interval, we should use technology (we use TI-83 plus). If software is not used, a t-distribution table is needed to find the p-value and t-critical value. Refer to your favorite statistics textbook for a t-table or use this t-table.
With a price differential of $13.64, we see that Avis is more expensive. Let’s confirm it with a one-sample t-test. To assess whether Budget is less expensive than Avis, we test the following hypotheses:
where is the mean difference in car rental prices (Avis minus Budget). The null hypothesis says that there is no difference in prices between Budget and Avis. The alternative hypothesis says that Budget is less expensive than Avis (i.e. Avis minus Budget > ).
The mean and standard deviation of the “differenced” sample (the last column in Table 1) are:
The one-sample t-statistic is:
The p-value of this t-test is found from the t-distribution with 34 degrees of freedom (one less than the sample size). There are two ways to accomplish this (looking up a table or using software). Based on a t-distribution table (such as this one), . Software (using TI-83 plus) gives a p-value that is much smaller, , which is approximately .
Because of the small p-value, the data provide clear evidence in favor of the alternative hypothesis (i.e. we reject the null hypothesis ). A price differential this large (as large as $13.64) is very unlikely to occur by chance if there is indeed no difference in prices between Budget and Avis. We now have evidence that Budget is less expensive on average (Avis is more expensive on average).
What is the magnitude of the price differential of Avis over Budget with a margin of error? We want to obtain a 95% confidence interval for the mean difference in car rental prices. To this end, we need the critical value from a t-distribution table. The margin of error is:
and the confidence interval is:
The estimated average price differential of Avis over Budget is $13.64 with a margin of error $5.02 with 95% confidence. On average, you tend to save anywhere from $8.62 to $18.66 for a one-day car rental if you go with Budget.
It is clear that Avis is more expensive (at least in terms of one-day rental of full size sedans on a Monday). Perhaps, other factors could alter the picture. For example, this comparison does not account for discount or special promotion. There are variations to the exercise done here. One is to compare week-long rentals. Another exercise is to compare vehicles in other classes (e.g. economy or SUV). Another one is to compare prices for busy seasons (e.g. holiday weekends).
- Moore. D. S., Essential Statistics, W. H. Freeman and Company, New York, 2010
- Moore. D. S., McCabe G. P., Craig B. A., Introduction to the Practice of Statistics, 6th ed., W. H. Freeman and Company, New York, 2009