Statistics Homework

1. Students at a major university are complaining of a serious housing crunch. Many of the university; s students, they complain, must commute too far to school because there is not enough housing near campus. The university officials respond with the following information: the mean distance commuted to school by students is 14.7 miles, and the standard deviation of the distance commuted is 3.3 miles.

Assuming that the university officials’ information is correct, complete the following statements about the distribution of commute distances for students at this university.

a. According to Chebyshev’s theorem, at least (what the percentage) _____ of the commute distances lie between 8.1 miles and 21.3 miles.

b. According to Chebyshev’s theorem, at least (what the percentage) _____of the commute distances lie between 4.8 miles and 24.6 miles.

c. Suppose that the distribution is bell-shaped. According to the empirical rule, approximately (percentage______) of the commute distance lie between 8.1 miles and 21.3 miles

d. Suppose that the distribution is bell shaped. According to the empirical rule, approximately 68% of the commute distance lie between __miles and __miles.

2. There is some evidence that, in the years 1981-85, a simple name change resulted in a short-term increase in the price of certain business firms; stocks (relative to the prices of similar stocks). (See D. Haorsky and P. Swyngedouw, “Does it pay to change your company’s name? A stock market perspective,” Marketing Science v.6, pp.320-35, 1987.)

Suppose that, to test the profitability of name changes in the more recent market ( the past five years), we analyze the stock prices of a large sample of corporations shortly after they changed names, and we find that the mean relative increase in stock price was about 0.77% with a standard deviation of 0.11%. Suppose that this mean and standard deviation apply to the population of all companies that changed names during the past 5 years. Complete the following statement about the distribution of relative increases in stock price for all companies that changed names during the past five years.

a. According to Chebyshev’s theorem, at least (percentage____) of the relative increases in stock price lie between 0.495% and 1.045%

b. According to Chebyshev’s theorem, at least (percentage______) of the relative increase in stock price lie between 0.55% and 0.99 %?

c. Suppose that the distribution is bell-shaped. According to the empirical rule, approximately 99.7% of the relative increases in stock price lie between ___percentage and _____percentage.

3. From the time of early studies by Sir Francis Galton in the late nineteenth century linking it with mental ability, the cranial capacity of the human skull has played an important role in arguments and IQ, racial differences, and evolution, sometimes with serious consequences. (see, for example, S.J. Gould, “The Mismeasure of Man, “1996)

Suppose that the mean cranial capacity measurement for modern, adult males in 1139 cc (cubic centimeters) and that the standard deviation is 217 cc. Complete the following statements about the distribution of cranial capacity measurements for modern, adult males.

a. According to Chebyshev’s theorem, at least (what percentage____of the measurement lie between 705 cc and 1573 cc

b. According to Chebyshev’s theorem, at least of 36% of the measurements lie between ____cc and ____cc (Round your answer to the nearest integer)

c. Suppose that the distribution is bell-shaped. According to the empirical rule, approximately 68% of the measurements lie between ___cc and ____cc.

d. Suppose that the distribution is bell-shaped. According to the empirical rule, approximately (what percentage___) of the measure ments lie between 705 cc and 1573 cc

4. Fill in the P (X = x) values in the table below to give a legitimate probability distribution for the discrete random variable X, whose possible vales are -1, 3, 4, 5 and 6

Value x and X	P (X-x)
-1	0.13
3	0.24
4
5	0.14
6

5. Fill in the P (X -x) values in the table below to give a legitimate probability distribution for the discrete random variable X, whose possible values are -1, 3, 4, 5, 6

Value x and X	P (X-x)
-1	0.12
3	0.26
4
5
6	0.27

6. Fill in the P (X = x) values in the table below to give a legitimate probability distribution for the discrete random variable X, whose possible values are 2, 3, 4, 5 and 6

Value x and X	P (X-x)
2	0.18
3	0.14
4
5	0.29
6

7. An ordinary (fair) coin is tossed 3 times. Out comes are thus triples of “heads” (h) and tails (t) which we write hth, ttt, etc. For each outcome, let R be the random variable counting the number of tails in each outcome. For example, if the outcome is thh, then R (thh) = 1. Suppose that the random variable X is defined in terms of R as follows: X = R-R²-2. The values of X are thus:

Outcome	hhh	hth	tth	hht	thh	htt	ttt	tht
Value of X	-2	-2	-4	-2	-2	-4	-8	-4

Calculate the probability distribution function of X, i.e. the function Px(ˣ). First, fill in the first row with the values of X. Then fill in the appropriate probabilities in the second row.

Value x of X
Px(ˣ)

8. Suppose that the genders of the three children of a certain family are soon to be revealed. Out comes are thus triples of “girls” (g) and boys (b), which we write gbg, bbb, etc. For each outcome, let R be the random variable counting the number of girls in each outcome. For example, if the outcome is gbg, then R (gbg) = 2. Suppose that the random variable X is defined in terms of R as follows: X = 2R² - 2R -1. The values of X are thus:

Outcome	bgg	gbg	bgb	ggg	bbb	ggb	bbg	gbb
Value of X	3	3	-1	11	-1	3	-1	-1

Calculate the probability distribution function of X, i.e. the function Px(ˣ). First, fill in the first row with the values of X. Then fill in the appropriate probabilities in the second row.

Value x of X
Px(ˣ)

9. An ordinary (fair) coin is tossed 3 times, outcomes are thus triple of “heads” (h) and tails (t) which we write hth, ttt, etc. For each outcome, let R be the random variable counting the number of heads in each outcome. For example, if the outcome is ttt, then R (ttt) =0. Suppose that the random variable X is defined in terms of R as follows: X = 3R-R² -2. The values of X are thus:

Outcome	hht	hhh	htt	ttt	thh	tth	tht	hth
Value of X	0	-2	0	-2	0	0	0	0

Calculate the probability distribution function of X, i.e. the function Px(ˣ). First, fill in the first two row with the values of X. Then fill in the appropriate probabilities in the second row.

Value x of X
Px(ˣ)

10. Let X be a random variable with the following probability distribution:

Value x of X	P (X =x)
10	0.05
20	0.25
30	0.15
40	0.40
50	0.15

Find the expectation E (X) and variance Var (X) of X.

a. E(x) =

b. Var (X) =

11. Let X be a random variable with the following probability distribution:

Value x of X	P(X=x)
30	0.05
40	0.35
50	0.20
60	0.40

Find the expectation E(X) and variance Var (X) of X

a. E(x) =

b. Var (X) =

12. Let X be a random variable with the following probability distribution

Value x of X	P (X=x)
0	0.05
1	0.40
2	0.10
3	0.10
4	0.35

Find the expectation E(X) and variance Var (X) of X

c. E(x) =

d. Var (X) =

13. The workers’ union at a certain university is quite strong. About 94% of all workers employed by the university belong to the worker’s union. Recently, the workers went on strike, and now a local TV station plans to interview a sample of 10 workers, chosen at random, to get their opinions on the strike.

a. Estimate the number of workers in the sample who are union members by giving the mean of the relevant distribution (that is, the expectation of the relevant random variable) Do not round the response

b. Quantify the uncertainty of your estimate by giving the standard deviation of the distribution. Round your response to at least three decimal places.

14. Suppose that we have decided to test Clara, who works at the Psychic Center, to see if she really has psychic abilities. While talking to her on the phone, we will thoroughly shuffle a standard deck of 52 cards (which is made up of 13 hearts, 13 spades, 13 diamonds, and 13 clubs) and draw one card at random. We will ask Clara to name the suit (heart, spade, diamond, or club) of the card we drew. After getting her guess, we will return the card to the deck, thoroughly shuffle the deck, draw another card, and get her guess for the suit of this second card. We will repeat this process until we have drawn a total of 15 cards and gotten her suite guesses for each.

Assume that Clara is not clairvoyant, that is assume that she randomly guesses on each card.

a. Estimate the number of cards in the sample for which Clara correctly guesses the suite by giving the means of the relevant distribution (that is, the expectation of the relevant random variable). Do not round your responses

b. Quantify the uncertainty of your estimate by giving the standard deviation of the distribution. Round your response to at least three decimals places.

15. A rainstorm in Portland, Oregon, wiped out the electricity in 5% of the households in the city. Suppose that a random sample of 70 Portland households is taken after the rainstorm.

a. Estimate the number of households in the sample that lost electricity by giving the mean of the relevant distribution (that is, the expectation of the relevant random variable). Do not round your response

b. Quantify the uncertainty of your estimate by giving the standard deviation of the distribution. Round your response to at least three decimals places.

16. The workers union at a university is quite strong. About 94% of all workers employed by the university belong to the worker union. Recently the workers went on strike, and now a local TV station plans to interview 4 workers (chosen at random) at the university to get their opinion on the strike. What is the probability that exactly 3 of the workers interviewed are union members? Round your response to at least three decimal places.

17. Suppose that 45% of all babies born in a hospital are boys. If 8 babies born in the hospital are randomly selected, what is the probability that fewer than 2 of them are boys?

a. Carry your intermediate computations to at least four decimals places, and round your answer to two decimals places.

18. The manufacturer of a fertilizer guarantees that, with the aid of the fertilizer, 70% of planted seeds with germinate. Suppose the manufacturer is correct. If 9 seeds planted with the fertilizer are randomly selected, what is the probability that more than 6 of them germinate? Carry your intermediate computations to at least four decimals places, and round your answer to two decimal places.

19. Suppose that the New England Colonials baseball team is equally likely to win a game at not to win it. If 4 colonials’ games are chosen at random, what is the probability that exactly 2 of those games are won by the colonials? Round your response to at least three decimal places.

20. From experience, an airline knows that only 85% of the passengers booked for a certain flight actually show up. If 6 passengers are randomly selected, find the probability that more than 4 of them show up. Carry your intermediate computations to at least four decimal places, and round your answer to two decimal places.