The F Distribution with 4 and 30 Degrees of Freedom (for a Regression Model with an Intercept Plus Four X Variables Tested over 35 Observations)

The F distribution is skewed to the right but tends toward normality as both numbers of degrees of freedom become very large.

Standard Deviation of B

where, once again, n is the number of observations (data points) and k is the number of estimated coefficients (intercept plus the number of slope coefficients). Notice that this t statistic measures the size of an individual coefficient estimate relative to the size of its underlying standard deviation.

This popular t statistic measures the size of the b coefficient relative to its standard deviation because both the size of b and its underlying stability are important in determining if, on average, b ^ 0. The t statistic measures the number of standard deviations between the estimated regression coefficient, b, and zero. If the calculated t statistic is greater than the relevant critical t value, taken from a table of values such as that found in Appendix C, the hypothesis that b = 0 can be rejected. Conversely, if the calculated t statistic is not greater than the critical t value, it is not possible to reject the b = 0 hypothesis. In that case, there is no evidence of a relation between Y and a given X variable.

Returning to the First National Bank example, the estimated coefficient for the number of new account applications X variable is 20.339. Given a standard deviation of only 1.85, the calculated t statistic = 10.99 > 3.169, the critical t value for n - k = 10 degrees of freedom at the a = 0.01 significance level. With 99 percent confidence, the hypothesis of no effect can be rejected. Alternatively, the probability of encountering such a large t statistic is less than 1 percent [hence the probability (p) value of 0.000 in Figure 3.4] when there is in fact no relation between the total costs Y variable and the number of new account applications X variable.

As a rough rule of thumb, assuming a large n > 30 sample size and a typical regression model of four or five independent X variables plus an intercept term, a calculated t statistic greater than two permits rejection of the hypothesis that there is no relation between the dependent Y variable and a given X variable at the a = 0.05 significance level (with 95 percent confidence). A calculated t statistic greater than three typically permits rejection of the hypothesis that there is no relation between the dependent Y variable and a given X variable at the a = 0.01 significance

MANAGERIAL APPLICATION 3.4

The personal computer revolution in business really got underway in the 1980s following the publication of powerful and easy-to-use spreadsheet software. Microsoft's Excel has blown away the original standard, Lotus 1-2-3, to make income statement and balance sheet analysis quick and easy. Recent versions incorporate a broad range of tools for analysis, including net present value, internal rate of return, linear programming, and regression. Such software also allows managers to analyze and display operating data using a wide variety of charting and graphing techniques. For basic statistical analysis, Excel features easy-to-use statistical capabilities like regression and correlation analysis.

For more detailed analysis, thousands of successful companies worldwide, including GE, 3M, and Ford Motor Company, use MINITAB statistical software. The latest version, MINITAB Release 13, is a complete stat package that makes statistical analysis easy and fast. For example, the Stat Guide is extremely helpful for interpreting statistical graphs and analyses. MINITAB Student software is a streamlined and economical version of Professional MINITAB, designed specially for introductory general and business statistics courses. The latest release of MINITAB Student features an intuitive and easy-to-use interface, clear manuals, and online help. MINITAB is a powerful programming language with sufficient documentation to help even novice users analyze data and interpret results.

For advanced statistical processing software, SPSS® 11.0 for Windows® embodies powerful statistical tools for in-depth analysis and modeling of business and economic data. SPSS® 11.0 for Windows® helps managers access data easily, quickly prepare data for analysis, analyze data thoroughly, and present results clearly. SPSS® 11.0 for Windows®is packed with online tutorials and plenty of examples to guide users, while interactive charts and tables help users understand and present their results effectively.

More than simply changing historical methods of data manipulation and analysis, this user-friendly software for the PC is fundamentally changing the way managers visualize and run their businesses.

See: For MINITAB software, see http://www.minitab.com; for SPSS products, see http://www.spss.com.

level (with 99 percent confidence). However, as described earlier, critical t values are adjusted upward when sample size is small in relation to the number of coefficients included in the regression model. In such instances, precise critical t values can be obtained from a t table, such as that found in Appendix C.

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