Economics 391 (Spring 2013) Professor Lamarche University of Kentucky Sample Midterm Exam Questions 1. Review Examples 5 8 and 10 in Lecture 1. 2. Review Examples 1 3 5 and 9 in Lecture 2. 3. Review Examples 1 and 3 in Lecture 3. 4. Review Examples 1 and 8 in Lecture 4. 5. Consider an investment portfolio of $50 000 in stock A and $50 000 in stock B. The expected value of A is 9.5% and B is 6%. The variance of A is 13% and the variance of B is 8%. The covariance between A and B is 18.6%. (a) Compute the portfolio’s weights associated with stock A and stock B. (b) Obtain the portfolio expected return. (c) Find the variance of the portfolio. 6. Approximately 80% of workers are sure that they will be able to retire at 65 years of age. Suppose 10 workers are randomly selected. (a) What is the probability that none of the workers will be able to retire at 65? (b) What is the probability that 2 workers will retire at 65? (c) Fin the expected value variance and standard deviation of the random variableX defined as ‘retirement at 65’. 7. In the U.S. it is estimated that an average person makes 4 visits a year to doctor’s o?ces. (a) What is the mean and standard deviation for an average person of the number of monthly visit to the doctor? Economics 391 (Spring 2013) Professor Lamarche University of Kentucky (b) What is the probability that an average person makes at least 1 monthly visit to the doctor? 8. A continuous random variable Y has P (Y > 10) = 0.16 and P (5? Y? 10) = 0.15. Find: (a) P (Y < 10) (b) P (Y < 5) (c) P (Y = 0) 9. Jenny will be receiving an antique item as a gift from her parents. She is not sure about the value of the item but she understands that she can sell the antique for $2 000 with 0.4 probability $200 with 0.5 probability and $20 with 0.10 probability. Would you suggest Jenny to accept $1 000 instead of receiving the antique? 10. The time between arrivals in an airport is exponentially distributed with a mean of 25 minutes. (a) What is the probability of no arrivals for more than one hour? (b) What is the probability of an arrival within 10 minutes? 11. Explain and describe (a) The di?erence between population mean and sample mean. (b) The concept of estimation biases. (c) The properties of the sampling distribution of the sample mean. 12. Suppose an investment is normally distributed with mean 10% and standard deviation 5%. Find the probability of not losing money. 13. Find the probability that the sample mean¯ is lower than 750. The distribution X of X is normal with mean 800 and variance 100. The sample mean is constructed using 25 observations. 14. Construct a 95% confidence interval estimator for the population mean ¯ = X 370.16 when ? = 75 and n = 25. Economics 391 (Spring 2013) Professor Lamarche University of Kentucky 15. According to the Statistical Abstract of the United States there are 31.1 mil-lions one-person households 38.6 millions two-person households 18.8 millions three-person households and 27.5 millions four or more person households. (a) Construct a table of each value of the random variable X the number of persons in the household and its estimated probability or frequency. (b) Compute and interpret P (X = 3) and P (X ? 4). (c) Find the mean variance and standard deviation for the population of the number of persons per household. 16. Consider the following Table: Y 0 1 2 3 P(Y) 0.10 0.30 0.40 (a) Complete the Table with the probability of P (X = 3). (b) Calculate the mean variance and standard deviation of Y . (c) Suppose now we have a new random variable X = 2 3Y . Construct a Table with the probability distribution of X for each value of Y and X. (d) Calculate the mean variance and standard deviation of X. (e) Use the properties of the expected value and variance to obtain the ex- pected value of X E(X) and the variance of X V (X). Compare your answers. 17. David is an insurance broker who believes that the probability of making a sale is 0.3. (a) Find the mean and variance of a random variable defined to be 1 it he makes a sale and 0 otherwise. (b) What is p and 1 ? p? Check that the probability distribution satisfies the two conditions discussed in class. (c) Suppose now David has 5 contracts and now the probability of making a sale is 0.4. Compute the probability that he makes: i. at most one sale Economics 391 (Spring 2013) Professor Lamarche University of Kentucky ii. between 2 and 4 sales. (d) Graph the distribution function checking that the probabilities are between 0 and 1 for allx and that the sum of the probabilities is equal to 1. 18. University H has decided to accommodate more students this Fall. Knowing that typically 40% of the students admitted actually enroll answer the following questions using Table 1 (Appendix B) in Keller 2012: (a) What is the probability that at most 6 students will enroll if the University H. o?ers admission to 10 new students? (b) If admission is o?ered to 20 students what is the probability that more than 12 students will enroll? (c) Assume that the probability increases to 70%. What is the probability that at least 12 out of 15 students will actually enroll? 19. Assume that Charlotte has two stocks: Disney (D) and Amazon (A). For each of these stocks the possible percent returns are{0% 5% 10% 15%}. The joint probability ofP (d a) = 0.0625 for all values ofD andA. (a) Construct a Table with joint probabilities. (b) Find the marginal probabilities. (c) Are Disney and Amazon stock returns independent? (d) What is the mean and variance of the stocksD andA? (e) What is the covariance betweenA andD? 20. The temperature in January is estimated to have a mean of 34 and a standard deviation of 6 in degrees Fahrenheit. Sally estimates that her heating bill can be predicted using the following formula: X= 300 ?5 × temp wheretemp is temperature. Find how much she is expected to pay in January. Moreover obtain the variance and standard deviation of the heating bill. 21. An instructor graded a large number of midterm exams and she considers that the test scores are normally distributed with a mean of 70 and a standard deviation of 10. Economics 391 (Spring 2013) Professor Lamarche University of Kentucky (a) What is the portion of students obtaining scores between 85 and 95? (b) What is the score needed to be at the top 10% of the class? 22. Response time at an online site can be modeled with an exponential distribution with a mean service of 5 minutes. John knows this and it is not sure to wait for an answer. What is the probability that for the reply to John’s request (a) What is the probability that the reply to John’s request takes longer than 10 minutes? (b) What is the probability that the reply to John’s request takes shorter than 10 minutes? (c) If the mean service is now 2 minutes what is the probability that it takes longer than 10 minutes? 23. James W. has a portfolio that includes 20 shares of Disney and 30 shares of Amazon. The price of Disney stock is normally distributed with a mean of 25 and a variance of 80. The price of Amazon is also normally distributed with a mean of 40 and a variance of 119. James finds out that these stocks are negatively correlated with? = ?0.4. (a) Find the mean and standard deviation of James W.’s portfolio. (b) Would you advice James to sell Amazon and buy Disney? (c) What is the probability that the value of the portfolio is greater than $2 000? (d) What is the mean and standard deviation of James’ portfolio if stocks are not correlated?
