Part 1- True/False: Answer A for True, answer B for False

1. For the events A and B, (A'∩ B')∩A = Ø

2. The probability of the union of two events is always greater than the sum of the individual probabilities of the events.

3. If A and B are independent, then A' and B' are independent.

4. The variance of the Poisson distribution is twice as large as the expected value.

5. If X is distributed with a binomial distribution, the Central Limit Theorem says that as the sample size of n increases, the distribution of X^- is normal.

6. For a random sample of size n of the variable X, the term (_i=1Σⁿ(X_i - X^-)²/n) is an unbiased estimater of the population variance σ_X².

7. A hypothesis test with P(Type I error)=0 will never reject the null hypothesis.

8. If the null hypothesis is false, a hypothesis test with P(Type I error)=0 will have Power=0.

9. In a linear regression, Σe_i = 0 only if r² =1.

10. In a linear regression, if Sx > 0 and b₁=0, then r²=0.

Part 2- Multiple Choice Problems:

The next three questions use information in the following paragraph.

You're playing Dungeons and Dragons, an old-school pen-and-paper role playing game. The game uses an 8-sided die, with the numbers 1-8 on each face. The probability of rolling each number is 1/8.

11. Let Y equal the sum of  the values from two independent rolls, then E[Y] is

a. 3.5

b. 4.5

c. 7.0

d. 7.5

e. 9.0

12. What is the variance of one throw of the die?

a. 4.75

b. 5.25

c. 5.60

d. 5.85

e. 6.10

13. Suppose the result of #12 was some number we'll call K. Then, what is the variance of Y, as defined in #11?

a. K

b. 2K

c. 3K

d. 4K

e. 5K

14. If Y is a continuous, uniformly distributed random variable over the interval (4,10), then the value of the PDF between 4 and 10 is:

a. 0

b. 0.067

c. 0.135

d. 0.167

e. 0.25

The next three questions use the following joint probability table.  Fill in the missing values.

15. What is the value of G?

a. .10

b. .15

c. .20

d. .25

e. .30

16. What is the value of P[S=400|A=10]?

a. .100

b. .125

c. .150

d. .175

e. .200

17. Calculate the expected value of A, conditional on S=300. That is, E[A|S=300]=

 a. 11.0

b. 11.5

c. 12.0

d. 13.0

e. 13.5

The next two problems utilize the information in the following paragraph.

Suppose there are two types of law school students: Short-haired and Long-haired. Additionally, law students generally choose between two types of firms when they graduate: Big and Small. Suppose the probability of being Short-haired is .8, and a Short-haired student will end up at a Big firm 90% of the time.

18. Then, what is the probability of being both Short-haired and working at a Small firm?

a. .08

b. .14

c. .55

d. .72

e. .40

19. Additionally, suppose the probability of working at a big firm is .75.  Then, amongst the Small firm students, the probability of being Long-haired is ____.

a. .17

b. .25

c. .40

d. .54

e. .68

20. On a given day in October, the number of customers that visit Hannah's Haunted House is normally distributed with a mean of 85 and a variance of 400. What is the probability that the number of customers on one day is between 75 and 100?

a. .3085

b. .4439

c. .4699

d. .5156

e. .5327

21. You are interested in estimating the average number of movies watched by Americans in a year. In a random sample of 25 individuals, you find the average to be 44.473 movies, with a sample variance of 81. What is the p-value for a two-tailed hypothesis test where the null is that the true population value is 40 movies?

a. .01

b. .02

c. .025

d. .05

e. .10

22. A car manufacturer claims that its vehicles average at least 28 miles per gallon. Suppose, in fact, the cars only get 26.5 miles per gallon on average, with a population variance of 1. For an appropriate hypothesis test at a .975 confidence level, what is the power of a test of the company's claim using a sample size of 4 cars?

a. .1492

b. .4658

c. .6543

d. .8508

e. .9236

23. Suppose amongst all the families on a Caribbean Cruise, the population variance of the number of children per family is 4. If we were to take a sample of 10 families, then the probability is .9 that the sample variance is less than ____.  (Hint: Use the χ² distribution.)

a. 6.53

b. 9.45

c. 14.7

d. 16.2

e. 17.8

24. In a linear regression, all else equal, which of the following will tend to make it more likely that we reject H₀: β₁=0?

a. Increasing the SSE.

b. Decreasing the absolute value of b₁.

c. Increasing the variance of X.

d. Decreasing the sample size.

e. Increasing the value of X^-.

The next six problems utilize information in the following paragraph.

A regression of height (X, in inches) on hourly earnings (Y, in dollars), yields the following results:

n=26

SSE=200

SST=1000

r=.447

S_X=3

ΣY_i=780

ΣX_i=1872

25. What is the value of Y^-?

a. 16

b. 22

c. 25

d. 28

e. 30

26. What is the value of b1 in the regression Y=b0+b1X?

a. 0.94

b. 0.98

c. 1.45

d. 2.26

e. 2.75

27. What is the variance of the regression? That is, what is the value of S_e²?

a. 28.2

b. 33.3

c. 35.6

d. 40.1

e. 41.6

Note: In order to avoid unnecessarily compounding any mistakes that are possible on the prior 3 questions, for the next 3 questions I am going to choose incorrect values for certain statistics as we continue the height/earnings problem. In particular, suppose that b₁=1.0 and the standard deviation of b₁ (that is, Sb1) is .7599. All other information from above continues to hold for the purposes of the remainder of the problem.

28. What is the predicted hourly earnings for someone 67 inches tall?

a. $21

b. $22

c. $23

d. $24

e. $25

 29. What is the p-value for a 2-tailed hypothesis test that β₁=0?

a. .01

b. .05

c. .10

d. .20

e. .25

30. The 99% confidence interval for β1 is 1±

a. 1.45

b. 1.65

c. 1.88

d. 1.93

e. 2.12