- Business Mathematics II, Sample Test 2: page 5 -
Business Mathematics II
Sample Midterm Examination 2
This test is only presented as a sample of the types of questions which might appear on an examination. The instructor of each section will set his or her own examinations, which may differ from the sample in both length and content.
Show all of the steps and all of the work that you use to solve each problem.
A list of formulas that you may find useful is provided on the last page of this test.
1. X is a continuous random variable, whose p.d.f., fX, is plotted below.
(i) Shade the region of the plot that corresponds to P(3 £ X £ 8).
(ii) Use the shaded region to estimate P(3 £ X £ 8). P(3 £ X £ 8) = ______.
2. X is a continuous random variable that takes values in the interval [0, 2]. X has p.d.f., fX(x) = -0.75×x2 + 1.5×x
(i) Set up, but do not evaluate, an integral that represents P(0.5 £ X).
(ii) Set up, but do not evaluate, an integral, that could be evaluated by Integrating.xls, for the expected value, mX, of X.
3. X is a finite random variable whose p.m.f. is given below.
x / 0 / 1 / 2fX(x) / 0.2 / 0.5 / 0.3
(i) Compute the mean, mX, of X.
(ii) Compute the variance, V(X), of X.
(iii) Compute the standard deviation, sX. of X.
4. The graph of the p.d.f., fX, for a continuous random variable, X, is shown below.
(i) Estimate the mean, mX, of X.
(ii) Estimate the standard deviation, sX. of X.
5. The following sample of size n = 4 was taken for the values of a random variable.
2, 2, 5, 3
(i) Compute the sample mean,
(ii) Compute the sample standard deviation, s.
6. Let X be the random variable the gives the number of sick leave days in the next year used by a randomly selected worker at your factory. The parameters for X are mX = 7 and sX = 2. Let be the random variable that is the mean of a random sample of size n = 100.
(i)
(ii)
(iii) Give a formula for the random variable, S, that is the standardization of .
(iv) What is the approximate distribution of S?
7. The graphs of four probability density functions are given below.
a. b.
c. d.
(i) Which one could correspond to a standard normal random variable? _____
(ii) Which one could correspond to a normal random variable with mX = 5 and sX = 1? _____
8. The ratios of monthly closing prices for two stocks, 3M and Lam Research have approximately normal distributions. Samples of size n = 60 for the ratios of each stock yield the follow sample statistics.
Stock / Sample mean / Sample Standard Deviation3M / 1.01 / 0.01
Lam Research / 1.01 / 0.03
(i) Use the available information to sketch the p.d.f. of the ratios for each stock. Hint: For 3M, the p.d.f. is approximately 40 at 1.01. For Lam Research, the p.d.f. is approximately 13 at 1.01.
3M Lam Research
(ii) For which stock is the sample mean more likely to provide a better estimate for the actual mean of the ratios? Explain you answer.
9. Let X be a normal random variable with mX = 12 and sX = 3. Fill in the information that would be needed to have the Excel function Random Number Generation create random values of X in Cells A1:A10.
10. You represent one of 10 companies that bid on oil leases. Historical data indicates that there will be a winner's curse of 15.6 million dollars in the auction process. Simulation shows a stable Nash equilibrium strategy of a 17.3 million dollar signal reduction.
(i) Suppose that all companies, yours included, bid their geologists' estimates for the fair value of a lease. If you win the bid, on average, how much extra profit would your company receive on the lease?
(ii) Suppose that all companies, yours included, bid their geologists' estimates for the fair value of a lease minus 17.3 million dollars. If you win the bid, on average, how much extra profit would your company receive on the lease?
(iii) In real world terms, what does it mean to say that "the Nash equilibrium strategy is to reduce a company's signal by 17.3 million dollars? In particular, what does equilibrium strategy mean?
Formulas
R(q) = q×D(q) P(q) = R(q) - C(q) MP(q) = MR(q) - MC(q)