120A Economics Professor Yongil Jeon Summer 2006
Name: _________________________ Student ID#: _________________________
Answer to Homework #2 Summer 2006 (Answer to Final Exam, Summer 2005)
Answer all questions on separate paper. This problem set should be handed in to Professor Jeon at the beginning of your review session on Tuesday, August 1, 2006. Problem sets may not be handed in once solutions have been distributed. Please write down your name and PID clearly. Good luck!
1-a)
(3 points) The Student t distribution is
a. the distribution of the sum of m squared independent standard normal random variables. b. the distribution of a random variable with a chi-squared distribution with m degrees of freedom, divided by m. c. always well approximated by the standard normal distribution. d. the distribution of the ratio of a standard normal random variable, divided by the square root of an independently distributed chi-squared random variable with m degrees of freedom divided by m. Answer: d 1-b) (3 points) When there are degrees of freedom (that is, with a large sample), the t distribution a. can no longer be calculated. b. equals the standard normal distribution. c. has a bell shape similar to that of the normal distribution, but with "fatter" tails. 2 d. equals the distribution. Answer: b
1-c)
(3 points) The central limit theorem a. states conditions under which a variable involving the sum of Y1 ,..., Yn i.i.d. variables becomes the standard normal distribution. b. postulates that the sample mean Y is a consistent estimator of the population mean Y . c. only holds in the presence of the law of large numbers. d. states conditions under which a variable involving the sum of Y1 ,..., Yn i.i.d. variables becomes the Student t distribution. Answer: a
2 1-d)
HOMEWORK #2, ECON 120A, Summer 2006 (3 points) The sample average is a random variable and a. b. c. d. is a single number and as a result cannot have a distribution. has a probability distribution called its sampling distribution. has a probability distribution called the standard normal distribution. has a probability distribution that is the same as for the Y1 ,..., Yn i.i.d. variables.
Answer: b
1-e)
^ (3 points) An estimator Y of the population value Y is more efficient when % compared to another estimator Y , if
a. b. c. d.
^ % E( Y ) > E( Y ). it has a smaller variance. its c.d.f. is flatter than that of the other estimator. ^ % both estimators are unbiased, and var( Y ) < var( Y ).
Answer: d
1-f)
(3 points) Degrees of freedom a. in the context of the sample variance formula means that estimating the mean uses up some of the information in the data. b. is something that certain undergraduate majors at your university/college other than economics seem to have an amount of. c. are (n-2) when replacing the population mean by the sample mean. 2 2 d. ensure that sY = Y . Answer: a
3
HOMEWORK #2, ECON 120A, Summer 2006
2. (12 points) Calculate the following probabilities using the standard normal distribution.
(a) (b) (c) (d) (e) (f) (g) (h) (i)
(1 point) Pr(Z < 0.0) (1 point) Pr(Z 1.0) (1 point) Pr(Z > 1.96) (1 point) Pr(Z < 2.0) (1 point) Pr(Z > 1.645) (1 point) Pr(Z > 1.645) (1 point) Pr(1.96 < Z < 1.96) (1 point) Pr(Z < -2.576 or Z > 2.576) (2 points) Pr(Z > z) = 0.10; find z.
(j)
(2 points) Pr(Z < -z or Z > z) = 0.05; find z.
Note that
. di norm(1.0) . di norm(1.96) . di norm(2.0) . di norm(1.645) . di norm(2.576) . di norm(1.2816)
.84134475 .9750021 .97724987 .95001509 .99500247 .9000085
Answer: (a) 0.5000; (b) 0.8413; (c) 0.0250; (d) 0.0228; (e) 0.0500; (f) 0.9500; (g) 0.9500; (h) 0.0100; (i) 1.2816; (j) 1.96.
4
HOMEWORK #2, ECON 120A, Summer 2006
3. (10 points) The height of male students at your college/university is normally
distributed with a mean of 70 inches and a standard deviation of 3.5 inches. If you had a list of telephone numbers for male students for the purpose of conducting a survey, what would be the probability of randomly calling one of these students whose height is (a) (2 points) taller than 6'0" (that is, 72 inches)?
(b)
(2 points) between 5'3" (that is, 63 inches) and 6'5" (that is, 77 inches)?
(c)
(2 points) shorter than 5'7" (that is, 67 inches), the mean height of female students?
(d)
(2 points) shorter than 5'0" (that is, 60 inches)?
(e)
(2 points)taller than Shaq O'Neal, the center of the L.A. Lakers, who is 7'1" tall (that is, 85 inches)?
Note that
. di norm(0.5714) .71613574 . di norm(2.0000) .97724987 . di norm(0.8571) .80430519 . di norm(2.8571) .99786234 . di norm(4.2857) .99999089 Answer: (a) Pr(Z > 0.5714) = 0.2839; (b) Pr( 2 < Z < 2) = 0.9545 or approximately 0.95; (c) Pr(Z < -0.8571) = 0.1957; (d) Pr(Z < -2.8571) = 0.0021; (e) Pr(Z > 4.2857) = 0.000009
5
4.
HOMEWORK #2, ECON 120A, Summer 2006
(15 points) Assume that two presidential candidates, call them Bush and Gore,
receive 50% of the votes in the population. You can model this situation as a Bernoulli trial, where Y is a random variable with success probability Pr(Y = 1) = p , and where Y = 1 if a person votes for Bush and Y = 0 otherwise. ^ Furthermore, let p be the fraction of successes (1s) in a sample, which is p(1 - p) distributed N(p, ) in reasonably large samples, say for n 40. n
Note that . di norm(1.26) .89616532; . di norm(2.00) .97724987
(a)
(5 points) Given your knowledge about the population, find the probability that in a random sample of 40, Bush would receive a share of 40% or less. Z < 0.40 - 0.50 = Pr (Z < -1.26) 0.104. In roughly ^ Answer: Pr ( p < 0.40) = Pr 0.25 40 every 10th sample of this size, Bush would receive a vote less of than 40%, although in truth, his share is 50%. (5 points) How would this situation change with a random sample of 100?
(b)
Z < 0.40 - 0.50 = Pr (Z < -2.00) 0.023. With this ^ Answer: Pr ( p < 0.40) = Pr 0.25 100 sample size, you would expect this to happen only every 50th sample. (c)
(3 points) Given your answers in (a) and (b), would you be comfortable to predict what the voting intentions for the entire population are if you did not know p but ^ had polled 10,000 individuals at random and calculated p ? Explain.
Answer: The answers in (a) and (b) suggest that for even moderate increases in the sample size, the estimator does not vary too much from the population mean. ^ Polling 10,000 individuals, the probability of finding a p of 0.48, for example, would be 0.00003. Unless the election was extremely close, which the 2000 election was, polls are quite accurate even for sample sizes of 2,500. (d)
(2 points) This result seems to hold whether you poll 10,000 people at random in the Netherlands or the United States, where the former has a population of less than 20 million people, while the United States is 15 times as populous. Why does the population size not come into play?
Answer: The distribution of sample means shrinks very quickly depending on the sample size, not the population size. Although at first this does not seem intuitive,
6
HOMEWORK #2, ECON 120A, Summer 2006
the standard error of an estimator is a value which indicates by how much the estimator varies around the population value. For large sample sizes, the sample mean typically is very close to the population mean. 5. (15 points) A population has a mean of 300 and a standard deviation of 30. A random sample of size 100 will be taken and the sample mean x will be used to estimate the population mean. (a) (3 points) What is the expected value of x ?
(b) (3 points) What is the standard deviation of x ?
(c) (3 points) Show the sampling distribution of x .
(d) (2 points) What does the sampling distribution of x show?
(e) (4 points) What is the probability that the sample mean will be within 3 of the population mean?
Note that
. di norm(-1) .15865525 . di norm(1) .84134475
(Answer) (a) The expected value of x is 300. 30 (b) The standard deviation of x is =3 100 (c) Normal with E( x )=300 and x =3. (d) The probability distribution of x -3 x - 3 ; therefore (e) - 3 ( x - ) 3 ; thus we have 3 x 3 P( - 1 Z 1 )=.8413-.1586=.6827
7
HOMEWORK #2, ECON 120A, Summer 2006
6. (15 points) U.S. News and World Report ranks colleges and universities annually. You randomly sample 100 of the national universities and liberal arts colleges from the year 2000 issue. The average cost, which includes tuition, fees, and room and board, is $23,571.49 with a standard deviation of $7,015.52. (a) (5 points) Based on this sample, construct a 95% confidence interval (the critical value = 1.96) of the average cost of attending a university/college in the United States. Answer: 23,571.49 1.96 7, 015.52 = 23,571.49 1.96*701.55 100 = (22,869.94, 24,273.04).
(b) (8 points) Cost varies by quite a bit. One of the reasons may be that some universities/colleges have a better reputation than others. U.S. News and World Report tries to measure this factor by asking university presidents and chief academic officers about the reputation of institutions. The ranking is from 1 ("marginal") to 5 ("distinguished"). You decide to split the sample according to whether the academic institution has a reputation of greater than 3.5 or not. For comparison, in 2000, Caltech had a reputation ranking of 4.7, Smith College had 4.5, and Auburn University had 3.1. This gives you the statistics shown in the accompanying table. Reputation Category Ranking > 3.5 Ranking 3.5 Average Cost (Y ) $29,311.31 $21,227.06 Standard Deviation of Cost ( sY ) $5,649.21 $6,133.38 N 29 71
Test the hypothesis that the average cost for all universities/colleges is the same independent of the reputation. What alternative hypothesis did you use? Answer: Assuming unequal population variances, t = = 6.33, 5, 649.212 6,133.382 + 29 71 which is statistically significant whether or not you use a one-sided or two-sided hypothesis test. Your prior expectation is that academic institutions with a higher reputation will charge more for attending, and hence a one-sided alternative would have been appropriate here. (29,311.31 - 21, 227.06)
(c) (2 points) What other factors should you consider before making a decision based on the data in (b)? Answer: There may be other variables which potentially have an effect on the cost of attending the academic institution. Some of these factors might be whether or not the college/university is private or public, its size, whether or not it has a religious affiliation, etc. It is only after controlling for these factors that the "pure" relationship between reputation and cost can be identified.
8
HOMEWORK #2, ECON 120A, Summer 2006
7. (15 points) Consider the following hypothesis test. H 0 : = 16 vs H 1 : 16 Data from a sample of six item are: 14, 16, 12, 15, 13, 14. (a) (3 points) Compute the sample mean.
(b) (3 points) Compute the sample deviation.
(c) (3 points) With = 0.05, what is the rejection rule?
(d) (3 points) Compute the value of the test statistics t.
(e) (3 points) What is your conclusion?
Note that . di invttail(5,0.025) 2.571
(ANSWER)
(a) x =
(b) s =
x
n
i
=
i
84 = 14 6 - x)
2
0 2 + 2 2 + 2 2 + 12 + 12 + 0 2 10 = = 1.41 6 -1 5 n -1 (c) with 5 degrees of freedom, t 5,0.025 = 2.571 =
(x
Reject H0 if t < -2.571 or t > 2.571
x - 14 - 16 = = -3.47 1.41 s 6 n (e) Reject H0, and thus conclude H1 is true.
(d) t =