statistics

Recitation 2 Problem Set Introduction to Econometrics – ECON 266 New York University Problem 1. Consider the joint probability (i) Compute the marginal probability distributions of X and Y . (ii) Compute the mean and variance of X and Y . (iii) Compute the covariance and correlations of X and Y . (iv) Are X and Y independent? (v) Compute the mean and variance of the W = 2X − Y Problem 2. George Allen has asked you to analyze his stock portfolio, which contains 10 shares of stock D and 5 shares of stock C. The joint probability distribution of the stock prices in shown in the Table below (e.g. the probability Stock C’s price is $50 and Stock D’s price is $40 is 0.05). Compute the mean and variance of the total value of his stock portfolio. (Hint: use 10 and 5 as weights when adding two random variables; if the question asked for the mean and variance of the portfolio “per stock,” you could use weights 10/15 and 5/15) 1 Problem 3. Headhunters locate candidates to fill vacant senior positions in companies. These placement companies are typically paid a percentage of the salary of the filled position. A placement company that specializes in biostatistics is considering a move into information technology (IT). It earns a fee of 15% of the starting salary for each person it places. Its numerous placements in biostatistics had an average starting salary of $125,000. Its first 50 placements in IT had an average starting salary of $140,000 (s = $20,000) but produced higher costs at the agency. If each placement in IT has cost the placement company $1,200 more than each placement in biostatistics, should the firm continue its push into the IT industry? (i) State the null and alternative hypotheses. Describe the parameters. (ii) Describe Type I and Type II errors in this context. (iii) Find the p-value of the test. Do the data supply enough evidence to reject the null hypothesis if α = 0.10? (Assume that the data meet the sample size condition.) Problem 4. Compute the a) test-statistic, b) p-value, and c) decision (by both comparing the test-statistic to the z-score/t-score and by comparing the p-value to the α level) for the following hypothesis tests: (i) n = 20, x̄ = 20.89, s = 3, α = 0.01 H0 : µ ≤ 20 Ha : µ > 20 (ii) n = 17, x̄ = 21.42, s = 5, α = 0.05 H0 : µ ≥ 24 Ha : µ < 24 (iii) n = 24, x̄ = 21.53, s = 2, α = 0.05 H0 : µ = 20 Ha : µ ̸= 20 Problem 5. Find the confidence interval for the following: (i) µ x̄ = 152, s = 35, n = 60, confidence level = 90%. (ii) µ x̄ = 8, s = 75, n = 25, confidence level = 95%. 2