## Info

Coefficient of determination = R2 = 83.3% Standard error of estimate = SEE = 14,875.95 units

Individual coefficients provide useful estimates of the expected marginal influence on demand following a one-unit change in each respective variable. However, they are only estimates. For example, it would be very unusual for a \$1 increase in price to cause exactly a -19,875.954-unit change in the quantity demanded. The actual effect could be more or less. For decision-making purposes, it would be helpful to know if the marginal influences suggested by the regression model are stable or instead tend to vary widely over the sample analyzed.

In general, if it is known with certainty that Y = a + bX, then a one-unit change in X will always lead to a b-unit change in Y. If b > 0, X and Y will be directly related; if b < 0, X and Y will be inversely related. If no relation at all holds between X and Y, then b = 0. Although the true parameter b is unobservable, its value is estimated by the regression coefficient b. If b = 10, a one-unit change in X will increase Y by 10 units. This effect may appear to be large, but it will be statistically significant only if it is stable over the entire sample. To be statistically reliable, b must be large relative to its degree of variation over the sample.

In a regression equation, there is a 68% probability that b lies in the interval b ± one standard error (or standard deviation) of the coefficient b. There is a 95% probability that b lies in the interval b ± two standard errors of the coefficient. There is a 99% probability that b is in the interval b ± three standard errors of the coefficient. When a coefficient is at least twice as large as its standard error, one can reject at the 95% confidence level the hypothesis that the true parameter b equals zero. This leaves only a 5% chance of concluding incorrectly that b ^ 0 when in fact b = 0. When a coefficient is at least three times as large as its standard error (standard deviation), the confidence level rises to 99% and the chance of error falls to 1%.

A significant relation between X and Y is typically indicated whenever a coefficient is at least twice as large as its standard error; significance is even more likely when a coefficient is at least three times as large as its standard error. The independent effect of each independent variable on sales is measured using a two-tail t statistic where t statistic = ———jb-

### Standard error of b

This t statistic is a measure of the number of standard errors between b and a hypothesized value of zero. If the sample used to estimate the regression parameters is large (for example, n > 30), the t statistic follows a normal distribution, and properties of a normal distribution can be used to make confidence statements concerning the statistical significance of b. Hence t = 1 implies

CASE STUDY (continued)

68% confidence, t = 2 implies 95% confidence, t = 3 implies 99% confidence, and so on. For small sample sizes (for example, df = n - k < 30), the t distribution deviates from a normal distribution, and a t table should be used for testing the significance of estimated regression parameters.

Another regression statistic, the standard error of the estimate (SEE), is used to predict values for the dependent variable given values for the various independent variables. Thus, it is helpful in determining a range within which one can predict values for the dependent variable with varying degrees of statistical confidence. Although the best estimate of the value for the dependent variable is Y, the value predicted by the regression equation, the standard error of the estimate can be used to determine just how accurate this prediction Y is likely to be. Assuming that the standard errors are normally distributed about the regression equation, there is a 68% probability that actual observations of the dependent variable Y will lie within the range Y ± one standard error of the estimate. The probability that an actual observation of Y will lie within two standard errors of its predicted value increases to 95%. There is a 99% chance that an actual observed value for Y will lie in the range Y ± three standard errors. Obviously, greater predictive accuracy is associated with smaller standard errors of the estimate.

San Francisco could forecast total unit demand, forecasting sales in each of the 30 market areas and then summing these area forecasts to obtain an estimate of total demand. Using the results from the demand estimation model and data from each individual market, it would also be possible to construct a confidence interval for total demand based on the standard error of the estimate.

A. Describe the statistical significance of each individual independent variable included in the San Francisco demand equation.

B. Interpret the coefficient of determination (R2) for the San Francisco demand equation.

C. What are expected unit sales and sales revenue in a typical market?

D. To illustrate use of the standard error of the estimate statistic, derive the 95% confidence interval for expected unit sales and total sales revenue in a typical market.