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Regression Analysis 135

Another measure, the sample standard deviation, s, is simply the square

root of the variance. The standard deviation is measured in the same units as

the sample observations. In the present example, we can compute the sample

variance to be s^2 11,706/16 731.6. In turn, the standard deviation of the

airline’s sales is given by:

With this data, how can we best predict next quarter’s sales, and how good

will this prediction be? We might use 87.2 seats, namely, the sample mean.

Given our data, this is probably better than any other estimate. Why?

Remember that in computing the sample variance, the squared differences

were measured from the mean, 87.2. But what if we had chosen some other

value, say 83 or 92, as our estimate? Computing the sum of squares around

either of these values, we find that it is much larger than around the mean. It

turns out that using the sample mean always minimizes the sum of squared

errors. In this sense, the sample mean is the most accurate estimate of sales. Of

course, there is a considerable chance of error in using 87.2 as next quarter’s

forecast. Next quarter’s sales are very likely to fluctuate above or below 87.2. As

a rough rule of thumb we can expect sales to be within two standard deviations

of the mean 95 percent of the time. In our example, this means we can expect

sales to be 87.2 plus or minus 54 seats (with a 5 percent chance that sales might

fall outside this range).

Let’s now try to improve our sales estimate by appealing to additional data.

We begin with the past record of the airline’s prices. These prices (quarterly

averages) are listed in the third column of Table 4.1. Again there is consider-

able variability. At high prices, the airline sold relatively few seats; when the air-

line cut prices, sales increased. Figure 4.1 provides a visual picture of the

relationship. Each of the 16 points represents a price-quantity pair for a par-

ticular quarter. The scatter of observations slopes downward: high prices gen-

erally imply low ticket sales, and vice versa.

The next step is to translate this scatter plot of points into a demand equa-

tion. A linear demand equation has the form

The left-hand variable (the one being predicted or explained) is called the

dependent variable. The right-hand variable (the one doing the explaining) is

called the independent (or explanatory) variable. As yet, the coefficients, a and

b, have been left unspecified (i.e., not given numerical values). The coefficient

a is called the constant term. The coefficient b (which we expect to have a neg-

ative sign) represents the slope of the demand equation. Up to this point, we

have selected the form of the equation (a linear one). We now can use regres-

sion analysis to compute numerical values of a and b and so specify the linear

equation that best fits the data.

The most common method of computing coefficients is called ordinary

least-squares(OLS) regression. To illustrate the method, let’s start by arbitrarily

QabP.

s 1 731.627.0 seats.

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