:solve the following questions.
A company examining its employee retention rates considers the number of complete
years a new employee works for a particular division before leaving. The results are
shown in the table below:
Division A
Division B
9
Division C
10 9
A company examining its employee retention rates considers the number of complete years a new employee works for a particular division before leaving. The results are shown in the table below: Division A 6 Division B Division C 8 10 5 7 (i) Perform a one-way analysis of variance at the 5% level to compare the retention rates for the three divisions. [5] (ii) Present the data in a simple diagram and hence comment briefly on the validity of the assumptions required for the analysis of variance. [2] (iii) Concern is expressed over the variability of the results in Division B. It is thought that this might be significantly different from the assumed common underlying variance. (a) Write down an estimate for the underlying common variance for all three divisions. (b) Calculate an unbiased estimate for the variance in Division B based on the data for Division B only. (c) Test at the 5% level whether the variance in Division B is significantly different from your estimate in (iii)(a). [5] [Total 12]5. (25 marks) (a) The following table gives the probability distribution of a random variable X x 0 1 2 -2 P(X=ac) a b 0.2 0.1 Assume the mean value of this distribution is E(X} = 0.5. (i) Find the values of a and b; [4 marks] (ii) Furthermore, nd the variance of this distribution Var(X). [4 marks] Assume that Bob is taking a quiz, which contains 6 questions. In this quiz, the rst two questions are True-False questions with 2 possible answers each, only one of which is correct. The last four questions are multiple choice questions with 4 possible answers each, only one of which is correct. Now assume Bob did not study for the quiz, so he guesses independently on each question. Find the probability that Bob answers exactly 1 question correctly in this quiz. (Hint: You may introduce two random variables X and Y. X is the number of correct questions to rst two True-False questions, and Y is the number of correct questions to the last four multiple choice questions.) [7 marks] In a quality control process, 2000 units are randomly selected each day from the production line to check for defects. It is known from past experience that the probability of having a defective unit is 0.05. If at any time, 110 or more defective units are detected, the process is stopped in that day. Suppose that this screening process is performed for 7 consecutive days. Find the (approximate) probability that the production process is stopped on exactly 2 days in this period. [10 marks] Question One (Budget constraints and perfect substitutes) 1. Suppose a consumer has an income of $500 and faces prices pr = 5 and pz = 10. a. Write the equation for the budget constraint. b. Draw the budget constraint, placing good Yon the horizontal axis. Label it BC. c. What is the slope of BC? d. Suppose income decreases to $300. Draw the new budget constraint and label it RS. 2. Confirm that if a consumer's utility function is described by U= 2.X+ Z, and prices are py = 2 and pz= 1, there is no unique utility maximizing solution regardless of income level. What does this tell you about X" and Z as commodities? (Hint: draw a graph showing a budget constraint and indifference curve using the information provided.) Question Two (Constrained consumer optimization) Linda loves buying shoes and going out to dance. Her utility function for pairs of shoes, S, and the number of times she goes dancing per month, 7, is U(S, )=2ST. (a) What are her marginal utility of shoes and her marginal utility of dancing? It costs Linda $50 to buy a new pair of shoes and $50 to spend a night out dancing. Assume that she has $500 to spend on clothing and dancing. (b) What is the equation for her budget line? Draw it (with 7 on the vertical axis), and label the slope and intercepts. (c) What is Linda's marginal rate of substitution? Explain what this tells us. (d) Solve mathematically for her optimal consumption bundle. Show in a diagram how to determine this bundle using indifference curves and using a budget line. Question Three (Demand curves - see examples 3.5 and 3.6 from Chapter 3 of Perloff, this is a hard question) Ryan has a constant elasticity of substitution utility function, U = x/ + x,, and the standard budget constraint discussed in lectures and the text. Derive Ryan's demand for both goods using the Lagrangian approach
