251y0322 10/27/03ECO251 QBA1
SECOND HOUR EXAM
March 21, 2003
Name: ____KEY______
Social Security Number: ______
Part I: (48 points) Do all the following: All questions are 2 points each except as marked. Exam is normed on 50 points including take-home.Please re-read, ‘Things that You should never do on an Exam or Anywhere Else, ‘ and especially recall that a probability cannot be above 1!
The following joint probability table shows the relation between two sets of events. Let event be that the individual is below 22 ( is over 21), and the event be that the individual had no traffic violations in the last 18 months, the event be that the individual has one traffic violation in the last 18 months and the event be that the individual has 2 traffic violations over the last 18 months. No individuals in this pool of drivers has more than 2 violations.
Note, to do the problems below, at least total the rows and columns -
- The probability that someone who is over 21 has no traffic violations is (To 2 decimal places):
a).63
b).60.
c).41
d)*.68 You have been asked for
e)None of the above.
- The probability that someone picked at random is over 21 and has no traffic violations is (To 2 decimal places):
a).63
b).60.
c)*.41 Joint probabilities are what the table shows!
d).68
e)None of the above.
- The probability that someone chosen at random is either under 22 or has 2 violations is:
a).40
b).06
c).46
d).47
e)*.41
f)None of the above
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- Which two events are independent?
a) and Note that ‘mutually exclusive’ and ‘independent are almost opposites.
b) and
c)* and . The definition of independence is . In this case and . But a better way to do this is to note that .
d) and
e) and
f)None of these.
- Which two events are mutually exclusive?
a)* and Complements are always mutually exclusive. None of the other pairs have a joint probability of zero.
b) and
c) and .
d) and
e) and
f)None of these.
In questions 6 and7 you need to know what , and are to do the problems. Show your work.
Solution:We can use the following table.
6.What is the probability that a person picked at random has at least one violation?
Solution: ‘For the 200th time, ‘at least one’ and ‘exactly one’ are rarely the same thing.
7.What is the mean and the standard deviation of the number of violations our drivers have?(6) 18
Solution:From the work above .
This can’t be and . These are a sample mean and variance, but there is no sample.
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8. Which of the following statements must be true if , and
(i) and are mutually exclusive. Explanation:
(ii) and are independent.Explanation:
(iii) and are collectively exhaustive.Explanation:
(iv) Explanation:
a)All are true but (i)
b)*All are true but (ii)
c)All are true but (iii)
d)All are true but (iv)
e)All are true
f)None are true.
9.According to a survey of American households, the probability that the residents own 2 cars if annual household income is over $25,000 is 80%. Of the households surveyed, 60% had incomes over $25,000 and 70% had 2 cars. The probability that the residents of a household own 2 cars and have an income less than or equal to $25,000 a year is (3):
a) 0.12.
b)0.18.
c)*0.22.
d)0.48.
Solution:The easiest way I know is to start with the given facts. Let ‘’ be the event that a household owns two cars, and ‘’ be the event that a family has an income over $25000. The problem says ,, and asks for
Using the second two facts we get . But 80% of the 60% of families that own two cars have incomes over 25000, that is So now we have If we fill in more, we get and, finally,. From the table we read
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10.According to a survey of American households, the probability that the residents own 2 cars if annual household income is over $25,000 is 80%. Of the households surveyed, 60% had incomes over $25,000 and 70% had 2 cars. The probability that annual household income is over $25,000 if the residents of a household own 2 cars is: (4)
a)0.42.
b)0.48.
c)0.50.
d)0.69. If we look at the table above . Or we could use Bayes’ rule to say
11. If you toss a coin 6 times, what is the chance that you get at least one head? (Show your work) (4)
Solution: The probability of at least one head is
12. If you have five pennies and four dimes in your pocket and you pick three coins, what is the chance that you get exactly 2 dimes? (Show your work) (4)
Solution:Remember that The number of ways you can get one penny from 5 is . The number of ways you can get two dimes from four is . The number of ways that you can get 3 coins from 9 is . our solution, which is an example of the Hypergeometric distribution is . We could also say that your chance of getting a dime on the first and second try is , your chance of getting a dime on the first and third try is and your chance of getting a dime on the second and third try is . Thus, your chance of getting exactly $0.20 is
13. Show, by using the appropriate formula, that the number of ways that you can pick 52 items from 55, if order is not important, is the same as the number of ways you can pick 3 items from 55.(4) and . These are obviously the same.
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TABLE 4-3
A survey is taken among customers of a fast-food restaurant to determine preference for hamburger or chicken. Of 200 respondents selected, 75 were children and 125 were adults. 120 preferred hamburger and 80 preferred chicken. 55 of the children preferred hamburger.
Let ‘ be hamburger, ‘’ be chicken, ‘’ be Adult and ‘’ be child. The numbers given are as follows. . Fill in the table and get or, if you divide by 200,
- Referring to Table 4-3, the probability that a randomly selected individual is a child and prefers chicken is ______.Solution:
- Referring to Table 4-3, the probability that a randomly selected individual is a child or prefers hamburger is ______.Solution: or
- Referring to Table 4-3, assume we know that a person has ordered chicken. The probability that this individual is an adult is ______.Solution: or
20. Assume that is a standardized random variable. What is the mean and standard deviation of ? Solution:Remember that and The outline says and ,Substitute -6 for , for and 3 for and we have and . The standard deviation is thus
251y0322 10/27/03 ECO251 QBA1
SECOND EXAM
March 21, 2003
TAKE HOME SECTION
-
Name: ____KEY ______
Social Security Number: ______
Throughout this exam show your work! Please indicate clearly what sections of the problem you are answering and what formulas you are using.
Part II. Do all the Following (9 Points) Show your work!
1. My Social Security Number is 265398248.
Take the following set of numbers:
10
23
17
16
32
35
44
45
33
21
and use your Social Security Number to provide the first digit of the first nine numbers, so that, for me, the numbers would become
210,623,517,316,932,835,244,445,833, 21. Compute a sample standard deviation for the resulting numbers. (3- 2point penalty for not doing)
Solution: Please recall the following.
, (Computational Formula) (Definitional formula)
So doing it both ways, we compute
1
Row
1 210 44100 -287.6 82714
2 623 388129 125.4 15725
3 517 267289 19.4 376
4 316 99856 -181.6 32979
5 932 868624 434.4 188703
6 835 697225 337.4 113839
7 244 59536 -253.6 64313
8 445 198025 -52.6 2767
9 833 693889 335.4 112493
10 21 441 476.6 227148
4976 3317114 0.0 841056
So
(A check on the validity of the mean.),(with a slight rounding error) and This means .
1
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2. At a local university 30% of women, 40% of men and 80% of monsters are business majors. 60% of students are women, 30% of students are men and 10% of students are monsters. (6)
a. What percent of students are business majors?
b. What percent of business majors are monsters?
c. List the following events - a person is a woman, - a person is a man, - a person is a monster, a person is a business major. Find , , , and show that your solution in b. satisfies Bayes’ rule.
Solution: According to the problem statement , and . Also
, and . If 100 students walk into a room, 60 will be women, 30 men and 10 monsters. We can show this as Of the 60 women, 30% or 18 will be business majors. Of the 30 men, 40% or 12 will be business majors. Of the 10 monsters, 80% or 8 will be business majors. We now have
We really ought to finish our table. or we might present the joint
probability table . This serves as a check that our probabilities
add to 1.
a) Thus 38 out of 100 or 38% are business majors.
b) 8 out of 38 business majors are monsters. or 21.05%.
c) From the work above: , and .
Since Bayes’ rule says we can either write and say or we can try as . Both of these are true except for rounding error.
1