Practice Problems If you can answer the following questions rather quickly, you should be well prepared for the exam. These problems cover most of the topics which will appear on exam 2 (chapters 10,11,12, and 13, in the Moore text).

1. The number of copies of the magazine Cosmopolitan that are sold daily at a convenience store is a random variable X which takes on the values 0,1,2,3,4 and 5. Most of the distribution of X is given in the table below.
   

   X	0	1	2	3	4	5
   P(X)	0.10	0.12	0.25	0.30	0.20	0.03

   1. What is the probability that X = 1?
   2. What is the probability that X is greater than or equal to 4?
   3. What is the probability that the store sells at least one copy of Cosmopolitan magazine on a randomly chosen day?
   4. Assuming daily sales are independent at this store, what is the probability that this store sells at least one copy of Cosmopolitan magazine for three days in a row?
2. Toot-Toots all you care to eat restaurant charges $8.95 per customer to eat at the restaurant. They find that their expense per customer (including the amount of food eaten and their expenses for labor), has a distribution that is noticably skewed to the right with a mean of $8.20 and a standard deviation of $3.
   1. Explain what the law of large numbers says about Toot-Toots customers and profits.
   2. If a couple (2 people entering the restaurant together) can be viewed as an SRS of size 2 from Toot-Toot's customer base, what are the mean and standard deviation of the sampling distribution of a couple's mean expense (that is, the average expense per customer, to Toot-Toot's, based on a sample of size 2)? Would it be safe to assume that the sampling distribution for a couple's mean expense has a Normal distribution? Explain.
   3. Assume that on a given day, 100 customers eat at Toot-Toots. If we view these 100 customers as an SRS from the customer base, what is the probability that Toot-Toots earns a profit on this day, i.e. what is P( ≤ 8.95)? What is the probability that Toot-Toots averages at least $0.50 profit per customer on this day, i.e. what is P( ≤ 8.45)?
3. For each setting below, decide if the random variable X has a binomial distribution. If X does have a binomial distribution, describe what a success is, what is n, what is p, and then calculate the mean and standard deviation of X. If it does not have a binomial distribution, explain why (which properties of the binomial setting are NOT satisfied?).
   1. Grandma Lumpit makes wonderful chocolate chip cookies, but her strength just isn't what it used to be, and she has trouble mixing the chips evenly in the dough. Some cookies have no chips, and others have as many as 15 chips. I select one of Grandma Lumpit's cookies at random and let X = the number of chips in the cookie.
   2. In fact, 7% of Grandma Lumpit's cookies have no chips. I select a dozen of her cookies at random (from a huge supply) and let X = the number of cookies with one or more chips.
   3. A collection of 3 dozen of Grandma Lumpit's cookies contains 3 cookies with no chips. I select 10 of these 36 cookies at random and let X = the number I get with no chips.
4. The department of transportation for Des Moines has found that 25% of all Des Moines parking tickets issued are NOT paid off within one month of their issue date. The department selects an SRS of 100 tickets issued one month ago and calculates X = the number that have not been paid off. The random variable X has a binomial distribution.
   1. For this setting, what is a success? What is n? What is p?
   2. What is the probability that X = 24?
   3. What is the probability that exactly 80 of the tickets HAVE been paid off?
   4. What are the mean and standard deviation of X?
   5. Is it safe to use the Normal approximation to the binomial for this random variable? Explain.
   6. What normalcdf command would approximate the probability that X is 30 or more (i.e. that X ≥30 )?
5. Three types of cereal, Lucky Charms, Trix, and Cinnamon Toast Crunch, are all giving away the same prizes in their boxes. One of the 6 prizes is an Underdog figurine that you  are very eager to get. It turns out that 15% of all Lucky Charms boxes, 20% of Trix boxes, and 12% of Cinnamon Toast Crunch boxes contain this Underdog figurine. You buy one box of each brand of cereal. Use U to denote that the Underdog figurine was in a box, and N to denote that a box did Not contain the Underdog figurine, and list the results in the order Lucky Charms, Trix, Cinnamon Toast Crunch (so for example, UNU denotes that the Lucky Charms box did contain the prize we want, the Trix box did not, and the Cinnamon Toast Crunch box did).
   1. There are 8 possible outcomes for this scenario (many of which are NOT equally likely), list them all.
   2. For each outcome, calculate the probability of that outcome, assuming the prizes in the boxes are independent of one another.
   3. Now let X denote the number of Underdog figurines that you end up with. What are the possible values for X?
   4. Find P(X = 2).
6. A survey of 479 children found that those who had slept with a nightlight, or in a fully lit room before the age of 2 had a higher incidence of nearsightedness (myopia) later in their childhood (Sacramento Bee, May 13, 1999). The raw data for each child consisted of two variables, each with three possible values. The first was whether they slept with lights (no lights, nightlight, or full light) and the second recorded their level of myopia (no myopia, myopia, or high levels of myopia). The data are reproduced below. Assume that the data related to this survey accurately reflects the probabilities for all "randomly chosen children".
   

   Slept with:	No Myopia	Myopia	High Myopia	Total
   Darkness	155	15	2	172
   Nightlight	153	72	7	232
   Full Light	34	36	5	75
   Total	342	123	14	479

   
   
   1. Is the variable related to whether the children slept with light categorical or quantitative?
   2. What is the probability that a randomly chosen child slept with a nightlight (not full light, just a nightlight)?
   3. What is the probability that a randomly chosen child developed myopia later in their childhood (not high myopia, just myopia)?
   4. What is the probability that a child slept with a nightlight or developed myopia later in their childhood?
   5. What is the probability that a child slept with a nightlight and developed myopia later in their childhood?
      
   6. What is the probability that a child developed myopia later in their childhood, given that they slept with a nightlight?
   7. Are the events "developed myopia" and "slept with a nightlight" independent?
      
      
      
7. In 1999, Elizabeth Dole was a candidate to become the first female president in US history, and many observers presumed that she would have particular strength among female voters. According to a Gallup poll, "She did slightly better among Republican women than among Republican men, but this strength was not nearly enough to enable her to challenge Bush. In the October poll, Dole received the vote of 16% of Republican women compared to 7% of Republican men." (www.gallup.com [2000]). At that time, the Republican party was 60% male and 40% female. What is the probability that a randomly selected Republican from the October poll would have voted for Dole?
8. Let X denote the number of brontosaurus burgers eaten by a randomly selected person at the Flintstone annual family reunion. Having organized this event for many years, Fred has determined that the mean of X is 1.4 brontosaurus burgers, and the standard deviation of X is 0.77 brontosaurus burgers. Wilma has noticed that the distribution of X is not very close to Normal, with a large number of people eating zero brontosaurus burgers, while some people (like Fred) eat an enormous number.
   1. If we select an SRS of 3 people at the Flintstone reunion, and calculate  = the mean number of brontosaurus burgers consumed by these 3 people, and let denote the sampling distribution of the sample mean (with SRSs of size 3),
      1. What is the mean of the sampling distribution ?
      2. What is the standard deviation of ?
      3. Why is it NOT safe to assume that the sampling distribution is Normal?
   2. If 300 people show up for the Flintstone reunion this year (and we can consider them as an SRS of size 300 from the population), what does the Law of Large Numbers tell us about the total number of brontosaurus burgers needed (about how many brontosaurus burgers would Fred need)?
   3. If 300 people (again assumed to be an SRS of size 300 from the population) show up at this year's Flintstone reunion, and Fred buys 450 brontosaurus burgers, what is the probability that Fred runs out of brontosaurus burgers? Hint convert the total of 450 to an value.
9. Julie is taking only 2 classes, English and History. Assume that her probabilities for getting As in her classes are 70% for English and 60% for History. Suppose also that there is a 20% chance that Julie does NOT get an A in either of these two classes.
   1. Make a Venn diagram for this situation.
   2. What is the probaility that Julie gets an A in both of her classes?
   3. What is the probability that Julie gets at least one A in her two classes?
   4. What is the probability that Julie gets an A in History, given that she got an A in English?
   5. Are the events "A in History class" and "A in English class" independent?