4.3 solutions

# There are 3 levels of salary at this company x 30000

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There are 3 levels of salary at this company. X 30,000 60,000 90,000 p(x) .45 .40 .15 12. What’s the mean salary? TRICK: change to units of thousands right away; calculations will be much easier! 30 x .45 + 60 x .40 + 90 x .15 = 51 (thousand) 13. What’s the variance of salary? E(X) 2 = 3060 (thousand) V(X) = E(X 2 ) – E(X) 2 = 3060 – 51 2 = 459 ( thousands squared) To make more sense, SD = = 21.42 (thousands) 14. Give everyone a \$5,000 raise. What’s the new average salary? E(X + 5000) = 51 + 5 = 56 (thousands) 2

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15. Give everyone in the salary example a 10% raise. What’s the new average salary? E(1.10X) = 1.1(51) = 56.1 (thousands) Everyone working at Bob’s diner makes \$8/hour. 16. What is the expected salary per hour? E(X) = 8 17. What is the variance of the salaries at Bob’s diner? V(X) = 0 Let X = price of stocks in Bob’s portfolio; let Y = price of stocks in Sue’s portfolio. Mean of X = \$20, SD = \$5; Mean of Y = \$30, SD = \$6 KNOW: E(X) = 20; SD(X0 = 5; V(X)=25; E(Y) = 30; SD(Y)=6; V(Y)=36 18. Suppose all the stocks in Bob’s portfolio double in price. a. What happens to the mean of Bob’s stocks? Want E(2X) = 2x20 = \$40 b. What happens to the variance? Want V(2X) = 4x25 = 100 19. Suppose all of Sue’s stocks each increase \$10.00 in price. a-b. What happens to the mean and variance of her stock prices? E(Y+10)=30+10=\$40; V(Y+10)=36 20. Assume Bob and Sue’s stocks are independent. a. What is the mean and standard deviation of their combined stock prices? E(X+Y)=20+30=\$50 V(X+Y) = 26+36=61 by independence; SD(X+Y)=square root of 61=\$7.8 b. What is the mean and standard deviation of the difference in their stock prices? E(X-Y)=20-30=\$-10 V(X-Y)=26+36=61 by independence; SD(X-Y)=\$7.8 21. Assume Bob and Sue’s stocks have correlation 0.4. a. What is the mean and standard deviation of their combined stock prices? E(X+Y)=\$50 as before V(X+Y)=25 + 36 + 2(.4)(5)(6)=85; SD(X+Y)=square root of 85 = 9.22 b. What is the mean and standard deviation of the difference in their stock prices? E(X-Y)=\$-10 as before V(X-Y)=25 + 36 - 2(.4)(5)(6)=37; SD(X-Y)=square root of 37=6.08 3
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