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ECON 83a: Statistics for Economic Analysis
Problem Set #3
1. The Car Repair Ratings website provides consumer reviews and ratings for garages in the United States and Canada. The time customers wait for service to be completed is one of the categories rated. The following table provides a summary of the wait-time ratings (1 = Slow/Delays; 10 = Quick/On Time) for 40 randomly selected garages located in the province of Ontario, Canada (Car Repair Ratings website, November 14, 2012).
Wait-Time Rating Number of Garages
1 6
2 2
3 3
4 2
5 5
6 2
7 4
8 5
9 5
10 6
a. Develop a probability distribution for x = wait-time rating.
b. Any garage that receives a wait-time rating of at least 9 is considered to provide outstanding service. If a consumer randomly selects one of the 40 garages for their next car service, what is the probability the garage selected will provide outstand- ingwait-time service?
c. What is the expected value and variance for x?
d. Suppose that 7 of the 40 garages reviewed were new car dealerships. Of the 7 new car dealerships, two were rated as providing outstanding wait-time service. Compare the likelihood of a new car dealership achieving an outstanding wait- time service rating as compared to other types of service providers.
2. The Knowles/Armitage (KA) group at Merrill Lynch advises clients on how to create a diversified investment portfolio. One of the investment alternatives they make available to clients is the All World Fund composed of global stocks with good dividend yields. One of their clients is interested in a portfolio consisting of investment in the All World Fund and a treasury bond fund. The expected percent return of an investment in the All World Fund is 7.80% with a standard deviation of 18.90%. The expected percent return of an investment in a treasury bond fund is 5.50% and the standard deviation is 4.60%. The covariance of an investment in the All World Fund with an investment in a treasury bond fund is –12.4.
a. Which of the funds would be considered the more risky? Why?
b. If KA recommends that the client invest 75% in the All World Fund and 25% in the treasury bond fund, what is the expected percent return and standard de-viation for such a portfolio? What would be the expected return and standard deviation, in dollars, for a client investing $10,000 in such a portfolio?
c. If KA recommends that the client invest 25% in the All World Fund and 75% in the treasury bond fund, what is the expected percent return and standard de- viation for such a portfolio? What would be the expected return and standard deviation, in dollars, for a client investing $10,000 in such a portfolio?
d. Which of the portfolios in parts (b) and (c) would you recommend for an aggres- sive investor? Which would you recommend for a conservative investor? Why?
3. The Pew Research Center surveyed adults who own/use the following tech- nologies: Internet,smartphone, email, and land-line phone (USA Today, March 26, 2014) and asked which of these technologies would be “very hard” to give up. The following responses were obtained: Internet 53%, smartphone 49%, email 36%, and land-line phone 28%.
a. If 20 adult Internet users are surveyed, what is the probability that 3 users will report that it would be very hard to give it up?
b. If 20 adults who own a land-line phone are surveyed, what is the probability that 5 or fewer will report that it would be very hard to give it up?
c. If 2000 owners of smartphones were surveyed, what is the expected number that will report that it would be very hard to give it up?
d. If 2000 users of email were surveyed, what is the expected number that will re- port that it would be very hard to give it up? What is the variance and standard deviation?
4. Many companies use a quality control technique called acceptance sampling to monitor incoming shipments of parts, raw materials, and so on. In the electronics industry, component parts are commonly shipped from suppliers in large lots. In- spection of a sample of n components can be viewed as the n trials of a binomial experiment. The outcome for each component tested (trial) will be that the com- ponent is classified as good or defective. Reynolds Electronics accepts a lot from a particular supplier if the defective components in the lot do not exceed 1%. Sup- pose a random sample of five items from a recent shipment is tested.
a. Assume that 1% of the shipment is defective. Compute the probability that no items in the sample are defective.
b. Assume that 1% of the shipment is defective. Compute the probability that ex- actly one item in the sample is defective.
c. What is the probability of observing one or more defective items in the sample if 1% of the shipment is defective?
d. Would you feel comfortable accepting the shipment if one item was found to be defective? Why or why not?
5. Cars arrive at a car wash randomly and independently; the probability of an ar- rival is the same for any two time intervals of equal length. The mean arrival rate is 15 cars per hour. What is the probability that 4 or more cars will arrive during any given hour of operation? Use the Poisson probability function to calculate this probability.