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251x0531 4/18/05

**ECO251 QBA1 **

**THIRD EXAM **

**Apr 25, 2005 **

Name: _____________________

Student Number: _____________________

Part I: 16 points.

*z*

follows the standardized Normal distribution

*z*

~

*N*

.

** Class Time (Circle)**

1pm 2pm

Find the following.

**Make diagrams! **

1.

*P*

2 .

63

*z*

2 .

63

2.

*P*

3 .

11

*z*

2 .

63

3.

*F*

*z*

3 .

11

4.

*z*

.

115

251x0531 4/18/05

*x*

follows the Normal distribution

*x*

~

*N*

2 .

63 , 2

.

Find the following.

**Make diagrams! **

5.

*P*

2 .

63

*x*

2 .

63

6.

*P*

3 .

11

*x*

2 .

63

7.

*F*

*x*

3 .

11

8.

*x*

.

115

2

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Part II: (15+ points) Do all the following: All questions are 2 points each except as marked. Exam is normed on 50 points including take-home. Most questions come from Wonnacott and Wonnacott (1990)

(

**Showing your work can give partial credit on some problems!**

**In open-ended questions it is expected. Please indicate clearly what sections of the problem you are answering and what formulas you are using. Neatness counts!**

) Remember that you may not be able to finish this section, so ration your time on each problem. [Numbers in brackets are a cumulative total]

1.

If

*y*

tends to decrease as

*x*

a) b) c) d)

1

1

.

0

0

.

.

.

increases, what can we say about the population correlation

?

2.

The riskiness of a portfolio made up of two investments

3.

a) b) c) d) will be higher when the covariance is zero. will be higher when the covariance is negative. will be higher when the covariance is positive. does not depend on the covariance. [2]

Seventy items are randomly selected from a pilot production run of

*N*

items to check their quality.

*x*

is the number of defective items in the sample. The distribution of

*x*

can be considered approximately binomial if

*N*

a) 15

is b) 15

2 c) d) e) f)

400

1500

All of the above

None of the above.

4.

There are 24 Million people living in California. Approximately 8 million are in Los Angeles. The

Census Bureau takes a random sample of 200 people from the state. Let

*x*

be the number of

5.

people in the sample who live in Los Angeles, then a) Binomial distribution with

*n*

24

*million x*

is most conveniently treated as having the

and

*p*

8

24

. b) Poisson distribution. c) Binomial distribution with

*n*

200 and

*p*

8

24

. d) Hypergeometric distribution e) none of the above.

Let

*x*

represent the number of times a fair die (6 sides) comes up with a 1 on top when the die is cast 300 times. What is a)

*E*

, b)

*Var*

(

*x*

), c)

*E*

**Show your work!**

(5) [11]

6.

Assume that a fair 6-sided die is cast a large number of times. What is the chance that the first time it comes up with a 1 on top is in the first 10 throws?

**Show your work!**

(2)

3

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7.

A student calculates that

*P*

9

*x*

a) Binomial Distribution

35

*F*

b) c) d) e) f)

Poisson Distribution.

Continuous Uniform Distribution

Geometric distribution

All of the above

None of the above.

*F*

. The student could be working with the

[15]

8.

If we are taking a sample of 3 from a population that is one-third defective, what is the probability that at least one item in our sample is defective if a)

*N*

30 ? and b)

*N*

is very large?

**Show your work!**

(4)

9.

(Extra credit) The time it takes to complete an exam has an exponential distribution with a mean of 40 minutes. Luce Cash is in a hurry to finish so she can go shopping. What is the probability that she will finish the exam in 30 minutes or less? What is the probability that she will still be working when the instructor collects the exams after 55 minutes?

4

251x0531 4/18/05

**ECO251 QBA1 **

**THIRD EXAM **

**Apr 25, 2005 **

**TAKE HOME SECTION**

**- **

Name: _________________________

Student 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 III. Do all the Following (19+ Points)

** Show your work!**

1. Before you start, personalize the data below as follows. Take the third to last digit of your student number and add it to the last digit of .06. Take half the third to last digit of your social security number and subtract it from each of the .09s.

A photographic processor is trying to determine the distribution of the times that the business can promise to customers.

*x*

is the time in days it takes the package to arrive and

*y*

is the time in days it takes the package to return to the customer. It takes 2 days to process the film, so that the total time is

*t*

*x*

*y*

2 days.

*y*

1

2

2

.10

.15

*x*

3

.15

.26

4

.05

.09

3 .05 .09 .06

For example Robin dePoore’s student number is 999799 so she changes the .06 to .06 + .07 = .13 and the two .09s to .09 - .035 = .055. This should not change the total sum of the numbers in the joint probability table. Find. a) The mean and standard deviation of both

*x*

and

*y*

. (2)

The covariance and correlation between

*x*

and

*y*

. (3) b) c) d) e) f)

Using only the results in a) and b), find the mean and variance of

Find the probability that the total time

*t*

*x*

*y*

2

*t*

(2).

is no more than 5 days (2).

Assume that because of an anthrax scare,

*x*

, the time it takes the package to arrive, doubles. Using only the numbers that you found in a) and b) and appropriate formulas, find the new covariance and correlation and the mean and variance of

*t*

Go back to the original values of

*x*

and

(2).

*y*

and assume that the marginal probabilities of

*x*

g) and

*y*

are correct, what would the joint probability table look like if

*x*

and

*y*

were independent?

(1)

Recall what the covariance of

*x*

variance of

*t*

*x*

*y*

2

and

be now? (1)

*y*

would be if they were independent. What would the

[13]

2. Personalize the data below by adding the third to last digit of your student number to 10.

*x y v*

7

*y*

9 .

Use only what you found

23 8

16

20

21

13

9

10 d) in b) and appropriate formulas.

If

*w*

2

*x*

3 and

*v*

7

*x*

9 , find the covariance and correlation between a) Using the computational formula, find

*w*

and

*v*

. Use only what you found in the sample variance of

*x*

. (2) b) and appropriate formulas. (Hint: b) Find the covariance and correlation between

*x*

and

*y*

.(2) figure out or find from problems what

*Cov*

is.

c) Find the covariance and correlation between

*w*

2

*x*

3 and

5