High-tech companies and stock options. Different kinds of companies compensate their key employees in different ways. Established companies may pay higher salaries, while new companies may offer stock options that will be valuable if the company succeeds. Do high-tech companies tend to offer stock options more often than other companies? One study looked at a random sample of 200 companies. Of these, 91 were listed in the Directory of Public High Technology Corporations, and 109 were not listed. Treat these two groups as SRSs of high-tech and non-high-tech companies. Seventy-three of the high-tech companies and 75 of the non-high-tech companies offered incentive stock options to key employees.6

(a) What proportion of the high-tech companies offer stock options to their key employees? What are the odds?

(b) What proportion of the non-high-tech companies offer stock options to their key employees? What are the odds?

(c) Find the odds ratio using the odds for the high-tech companies in the numerator. Describe the result in a few sentences.

(d) Find the log odds for the high-tech ?rms. Do the same for the non-high-tech ?rms.

(e) De?ne an explanatory variable x to have the value 1 for high-tech ?rms and 0 for non-high-tech ?rms. For the logistic model, we set the log odds equal to β0 + β1 x. Find the estimates b0 and b1 for the parameters β0 and β1 .

(f) Show that the odds ratio is equal to eb1 .