Question

Question 2 – Key Incentives (4 questions – 10 marks)

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.

a) [5 marks] Test whether these two types of companies are equally likely to offer this kind of benefit to their employees. Use the critical value approach and a 0.05 level of significance. Perform the test manually.

b) [1 mark] Explain how to find the p-value manually (i.e., indicate what probability has to be calculated).

c) [2 marks] Finally, compute manually the 95% 2-sided confidence interval for the true difference between the proportions of high-tech and non-high-tech companies that offer incentive stock options to key employees.

d) [2 marks] Explain how the results you obtained in parts b) and c) are consistent with your conclusion in part a).

Answer 1

a) [5 marks] Test whether these two types of companies are equally likely to offer this kind of benefit to their employees. Use the critical value approach and a 0.05 level of significance. Perform the test manually.

The hypothesis being tested is:

H₀: p₁ = p₂

H_a: p₁ > p₂

The test statistic, z, is 1.83.

The critical z value is 1.645.

Since 1.83 > 1.645, we can reject the null hypothesis.

Therefore, we can conclude that high-tech companies tend to offer stock options more often than other companies.

b) [1 mark] Explain how to find the p-value manually (i.e., indicate what probability has to be calculated).

The p-value is 0.0335.

Since the p-value (0.0335) is less than the significance level (0.05), we can reject the null hypothesis.

Therefore, we can conclude that high-tech companies tend to offer stock options more often than other companies.

c) [2 marks] Finally, compute manually the 95% 2-sided confidence interval for the true difference between the proportions of high-tech and non-high-tech companies that offer incentive stock options to key employees.

The 95% 2-sided confidence interval for the true difference between the proportions of high-tech and non-high-tech companies that offer incentive stock options to key employees is between -0.0053 and 0.2335.

d) [2 marks] Explain how the results you obtained in parts b) and c) are consistent with your conclusion in part a).

Since the confidence interval contains 0, we can conclude that high-tech companies tend to offer stock options more often than other companies.

p1	p2	pc
0.8022	0.6881	0.74	p (as decimal)
73/91	75/109	148/200	p (as fraction)
73.	75.	148.	X
91	109	200	n
	0.1141	difference
	0.	hypothesized difference
	0.0623	std. error
	1.83	z
	.0335	p-value (one-tailed, upper)
	1.645	critical z
	-0.0053	confidence interval 95.% lower
	0.2335	confidence interval 95.% upper
	0.1194	margin of error