1. Suppose that X is a random variable with mean 20 and standard deviation 5. Also suppose that Y is a random variable with mean 40 and standard deviation 10. Assume that the correlation between X and Y is 0.5. Find the mean and variance of the random variable Z = 3X + 2Y .

2. Nonstandard dice can produce interesting distributions of outcomes. You have two balanced, six-sided dice. One is a standard die, with faces having 1, 2, 3, 4, 5, and 6 spots. The other die has two faces with 1 spot, two faces with 3 spots, and two faces with 6 spots. Find the probability distribution for the total number of spots Y on the up-faces when you roll these two dice.

3. A carnival game offers a $100 cash prize for anyone who can burst the balloon by throwing a dart. It costs $5 to play, and you're willing to spend up to $20 trying to win. You estimate that you have about a 10% chance of hitting the balloon on any throw. A winning is the profit you make taking into account the cash prize and the amount paid to play the game. (a) Create a probability model for this carnival game. (b) Find the expected number of darts you'll throw. (c) Find your expected winnings. Note: Assume that you stop playing after winning the game.

4. Explain what is wrong in each of the following statements. (a) The central limit theorem states that for large n, the population mean is approximately Normal. (b) For large n, the distribution of observed values will be approximately Normal. (c) For sufficiently large n, the empirical rule rule says that X should be within 2 about 95% of the time. 5. A high-speed packing machine can be set to deliver between 11 and 13 ounces of a liquid. For any delivery setting in this range the amount delivered is normally distributed with some mean amount and with standard deviation 0.08 ounce. To calibrate the machine it is set to deliver a particular amount, many containers are filled, and 25 containers are randomly selected and the amount they contain is measured. Find the probability that the sample mean will be within 0.05 ounce of the actual mean amount being delivered to all containers.

6. A tablet PC contains 3217 music files. The distribution of file size is highly skewed with many small file sizes. Assume that the standard deviation for this population is 3.25 megabytes (MB). (a) What is the standard deviation of the average file size when you take an SRS of 25 files from this population? (b) What is the probability of getting an average file size greater than 15 MB for an SRS of 25 files if the average of all 3217 files is 14 MB? (c) How many files (n) would you need to sample if you wanted the standard deviation of the average file size to be no larger than 0.50 MB?

7. Should you use the binomial distribution? In each of the following situations, is it reasonable to use a binomial distribution for the random variable X? Give reasons for your answer in each case. If a binomial distribution applies, give the values of n and p. (a) You toss a fair coin until a tail appears. X is the count of the number of tosses that you make. (b) Most calls made at random by sample surveys don't succeed in talking with a person. Of all calls to New York City, only one-twelfth succeed. A survey calls 500 randomly selected numbers in New York City. X is the number of times that a person is reached. (c) You deal 10 cards from a shuffled deck of standard playing cards and count the number X of black cards.

8. A particular telephone number is used to receive both voice calls and fax messages. Suppose that 25% of the incoming calls involve fax messages, and consider a sample of 5 incoming calls. (a) (2 points) What is the probability that at most 2 of the calls involve a fax message? (b) (2 points) What is the expected number and standard deviation of calls among the 5 that involve a fax message? 2