Question 1:
Every year, a local university publishes the average expenditures to give international students financial information when studying in New Zealand. The data provided in the file expenses.csv come from a survey of 20 single men and 20 single women. The units of expenditure are New Zealand dollars and divided into four categories:
housing housing, including electricity and water
food foodstuff, including alcohol and tobacco
goods other goods, including clothing, footwear, and durable goods,
service services, including movies, transport and vehicles
Use appropriate graphical methods to determine how money being spent between the four categories depends on total expenditure and whether this relationship differs for men and women. State your conclusions.
Question 3:
Consider the following game. A box contains a fair coin A and a two-headed coin B. A coin is selected at random and tossed twice. If head appears both times, the player wins $10. If it is a head and a tail (or the other way around), the player wins $5. But if tail appears both times, the player must pay $10. Let R denote the reward obtained when playing this game.
a. Represent this problem in an event tree.
b. If heads appear both times, find the probability p that the coin is two - headed.
c. Compute E[R].
d. Compute Var[R].
Suppose that you play this game 5 times. Let X denote the number of times that you win money. (Hint: For questions e - g, calculate the probability of winning first.)
e. What is the probability that you win money exactly 3 times?
f. What is the probability that you win money more times than you lose money?
g. What is the expected number of times that you will win money, E[X]?
Question 4
A kitchen appliance manufacturer has a new product line called NuKitch, developed to last longer and more energy efficient. The company plans to extend their standard one - year replacement warranty to 3.5 years of one of its products, which is the espresso machine. After extensive testing, under normal domestic use, it's lifespan is normally distributed with a mean of seven years and a standard deviation of two. The "life" of the espresso machine ends when it fails and needs to be repaired or replaced.
a. What is the probability that the product will last more than 5 years before it fails?
b. What is the probability that the lifespan of the product is less than 4 years or more than 7 years when it fails?
c. How many years before 15% of this product are no longer working?
d. Suppose that 10 espresso machines are selected at random. What is the probability that 4 of them will last less than 5 years before it fails?
Question 5
A local department store packages ground beef using meat trays of two sizes - small, which can hold a pound of meat and medium, which can hold around 3 pounds of meat. Thirty - five (35) packages in the smaller tray were randomly weighed and produced an average of 1.01 pounds and a standard deviation of 0.18 pound.
a. Construct a 95% confidence interval for the average weight sold in the smaller trays.
b. Explain the meaning of the phrase "95% confident" in this situation.
c. Should the confidence interval in part a concern the quality control department if the quality control department intends that the amount of ground beef in the smaller trays should be 1 pound on average? Explain.