C22.0103 Homework Set 3 Spring 2012
N N N N N N N N N N N N N N N N N N N N N N N N N N
1. The file CDCLifeTable is available from the course web site. It’s in the M folder; the direct link is
http://people.stern.nyu.edu/gsimon/statdata/B01.1305/M/index.htm
This has data for non-Hispanic whites, with gender corresponding to separate sets of columns.
a. Plot the female and male yearly hazards (the q’s) as a function of age. These should be overlaid on a single graph.
HINT: In using Minitab, select Graph Scatterplot, and then use “with connect line.” Set up the information panel like this:
Use Multiple Graphs to put both graphs on the same axis. You can edit the labels if you wish by double-clicking on graph features.
b. Compute the gender ratio as a function of age. Let’s say you choose to do this as “females per 100 males.” Plot this as a function of age.
HINT: In Minitab, use Calc Calculator.
c. Consider the marriage of Joseph, age 25, to Felicia, also age 25. Assume that their future survival mechanisms operate independently (even though this is a questionable assumption for a married couple). Find the probability that
both are alive at age 75,
Joseph will be alive at age 75 but Felicia will have died
Felicia will be alive at age 75 but Joseph will have died
both die before age 75
2. Consider the following life table:
Age (years) / Expectation of Future Life (years)Birth to 20 / 30
20 to 25 / 27
25 to 30 / 25
30 to 35 / 22
35 to 40 / 20
40 to 41 / 19
41 to 42 / 18
42 to 43 / 17
43 to 44 / 16
44 to 45 / 15
45 to 46 / 14
46 to 47 / 13
47 to 48 / 12
48 to 49 / 11
49 to 50 / 10
50 to 55 / 9
55 to 60 / 7
60 and up / 5
This table is presented in terms of expected future lifetime, rather than the more common (and more useful) x and dx notation. The Age column is given as intervals, but it’s easy to link to the more conventional notation. For instance, think of the row “42 to 43” as simply being “42.” This row refers to someone who has reached the 42nd birthday but not the 43rd birthday.
Make a list of things that are wrong with this table. You need not do elaborate arithmetic to form this list.
This is a genuine life table, and it was actually used!
3. Suppose that a pizza shop offers ten different toppings for its pizzas, and you select three of these at random.
a. What is the probability that mushrooms will be among your selections?
b. What is the probability that both mushrooms and onions will be among your selections?
c. What is the probability that your selections will be mushrooms, onions, and black olives?
d. How exactly would you go about selecting three toppings at random?
4. Suppose that you are dealt five cards, at random, from a standard deck.
a. What is the probability that exactly four of these cards will be spades?
b. What is the probability that at least four of these cards will be spades?
5. In games of the KENO or LOTTO style, the bettor selects numbers from a fixed set. Then the game operator selects another set of numbers, and the bettor wins according to the number of matches.
a. Suppose that the game uses the numbers 1 through 50, and suppose that the operator selects eight of these. If the bettor selects five numbers, find the probability that there are exactly five matches. HINT: You might find the number = 536,878,650 to be useful.
b. Suppose that the game uses the numbers 1 through 50, and suppose that the operator selects ten of these. If the bettor selects five numbers, find the probability that there are exactly five matches. Also note whether this probability is larger or smaller than the probability in a. HINT: You will want the number =10,272,278,170.
c. Suppose that the game uses the numbers 1 through 50, and suppose that the operator selects ten of these. If the bettor selects six numbers, find the probability that there are exactly six matches. Note whether the probability here is larger or smaller than the probability in b.
HINT: You can do these with a calculator, of course, but the calculations call be done more easily with Minitab’s hypergeometric option. The command is Calc Þ Probability Distributions Þ Hypergeometric.
6. The random variable Y has the probability distribution described in this table:
y / 1 / 2 / 3 / 4P[ Y = y ] / 0.30 / 0.40 / 0.20 / 0.10
Find the mean and standard deviation of the random variable Y.
7. The random variable Q has the probability distribution described in this table:
q / 100 / 200 / 300 / 400P[ Q = q ] / 0.20 / 0.50 / 0.20 / 0.10
Find the mean and standard deviation of Q. (After problem 6, this should be very easy.)
8. The random variable X has this distribution:x / 37 / 38 / 39 / 40 / 41
P[ X = x ] / 0.06 / 0.25 / 0.38 / 0.25 / 0.06
Find the mean EX.
9. Suppose that you are playing a game of chance in which the probability of winning is0.14. What is the probability that you’ll win exactly once in six turns?
10. The admissions office of a small, selective liberal-arts college will only offer admission to applicants who have a certain mix of accomplishments, including a combined SAT score of 1,400 or more. Based on past records, the head of admissions feels that the probability is 0.42 that an admitted applicant will come to the college. If 600 applicants are admitted, what is the probability that 250 or more will come? Note that “250 or more” means the set of values {250, 251, 252, …, 600}.
11. Consider the admissions office in the previous problem. Based on financial considerations, the college would like a class size of 250 or more. Find the smallest n, number of people to admit, for which the probability of getting 250 or more to come to the college is at least 0.95.
HINT 1: Based on your solution to the previous problem, you see that n=600 is not large enough.
HINT 2: You will have to proceed by trial and error.
HINT 3: You want P[ X³250 ] to be 0.95 or larger. The organization of Minitab makes it easier to ask for P[ X £249 ] to be 0.05 or less.
12. If X is the binomial phenomenon based on n trials and with success probability p, then EX = np and SD(X) = standard deviation of X = . In the situation of problems 8 and 9, with n = 650 and p = 0.42, find the mean and standard deviation of the number of students who will come to the college.
14. Assume that P(A) = 0.7 and P(B) = 0.2.a. If A and B are mutually exclusive, find P(AÇ B).
b. If A and B are mutually exclusive, find P(AÈ B).
c. If A and B are independent, find P(AÇ B).
d. If A and B are independent, find P(A È B).
15. Andrea and Betty are making a wager over a sequence of coin tosses. Andrea wins the first time there are two consecutive heads. Betty wins the first time tails-heads are observed consecutively.
For example, these sequences are wins for Betty:T H
T T T H
T T T T T H
What is the probability that Andrea wins this game?
16. The Sweet Novelty Company offers the special expensive ($150 each) Chocolate Groundhog product. During January, the Internet orders for this very expensive item come in at a rate of 1.2 per day, and it is believed that this phenomenon can be described as a Poisson random variable with l = 1.2. Find
a. the probability that there will be three or more orders on any day.
b. the probability that, in a five-day work week, there will be at least one day on which there are three or more order.
1
N Ógs2012