Question

University Endowments (Questions 14 16) Private colleges and universities rely on money contributed by individuals and corporations for their operating expenses. Much of this money is invested in a fund called an endowment, and the college spends only the interest earned by the fund. A recent survey of eight private colleges revealed the following endowments (in millions of dollars): 75.1, 53, 249.9,497.2, 114.4,167.8,110.1, and 224.8. 14. What value will you used as an unbiased point estimate for the standard deviation of endowment of all private colleges a. 143.247 b. 133.995 d. 444.2 c. 141.1 15. Compute the p-value to test the hypothesis that the true population variance of Endowments made to private universities is different from 19600 at 5% significance level. b. 0.605 c. 0.395 a. 0.041 d. 0.791 16. From Question 15 above, state your conclusion We reject the null and conclude that the true variance is significantly different from 19600 We failed to reject the null and conclude that the true variance is significantly different from 19600 We reject the null and conclude that the true variance is not significantly different from 19600 We failed to reject the null and conclude that we do not have enough evidence to suggest that the true variance is significantly different from 19600 a. b. c. d.

Answer 1

$14.~Unbiased~point~estimate~for~the~standard~deviation~of~endowment~of\\ all~private~colleges=s=\sqrt{\frac{1}{n-1}\sum_{i=1}^n(x_i-\bar{x})^2}=143.247,~n=8.\\ where,~\bar{x}=\frac{1}{n}\sum_{i=1}^nx_i=186.5375\\ Option~a.\\ 15.~Null~hypothesis,~H_0:\sigma^2=19600~\\vs.~Alternative~hypothesis,~H_1:\sigma^2\neq 19600\\ where,~\sigma^2=true~population~variance\\ Test~statistic=\chi^2=\frac{(n-1)s^2}{19600}=7.3285\\ p-value=2P(\chi^2>7.3285|\chi^2\sim \chi^2_7)=0.791.\\ Option~d.\\ Since~p-value>0.05~so\\ Option~d.~We~failed~to~reject~the~null~and~conclude~that~we~do~not~have~\\~enough~evidence~to~suggest~that~the~true~variance~is~significantly~different~from~19600.$