Statistics Homework

1. 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 about IQ, racial differences, and evolution, sometimes with serious consequence. (See, for example, S.J. Gould, “The Mis measure of Man, “1996)

Suppose that the mean cranial capacity measurement for modern, adult males is 1171 cc (cubic centimeters) and that the standard deviation is 283 cc. Complete the following statements about the distribution of cranial capacity measurement for modern, adult males.

(a) According to Chebyshev’s theorem, at least 36% of the measure lie between_____cc and ______cc. (Round your answer to the nearest integer.)

(b) According to the Chebyshev’s theorem, at least (percentage) _____of the measurements lie between 605 cc and 1737 cc.

(c) Suppose that the distribution is bell-shaped. According to the empirical rule, approximately (what percentage) _____of the measurements lie between 605 cc and 1737 cc.

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

2. Find 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, and 6

Value x of X	P (X = x
1	0.13
3
4	0.24
5
6	0.25

3. An ordinary (fair) coin is tossed 3 times. Outcomes are thus triples of “head” (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 tht, then R (tht)=2. Suppose that the random variable X is defined in terms of R as follows: X=4R-2R²-2. The value of X are thus:

Outcome	hth	ttt	tht	hhh	thh	tth	htt	hht
Value of X	0	-8	-2	-2	0	-2	-2	0

Calculate the probability distribution function of X, i.e. the function pX(x). The 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)

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

Value x of X	P (X= x)
-1	0.15
0	0.10
1	0.05
2	0.10
3	0.60

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

E(x) =

Var (X) =

5. A machine that manufactures automobile pistons is estimated to produce a defective piston 3% of the time. Suppose that this estimate is correct and that random sample of 90 pistons produced by this machine is taken.

a. Estimate the number of pistons in the sample that are defective 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 decimal places.

Answer: A

B

6. Anita’s a fast food chain specializing in hot dogs and garlic fries, keeps track of the proportion of it customers who decide to eat in the restaurant (are opposed to ordering the food “ to go”), so it can make decisions regarding the possible construction of in store play areas, the attendance of it’s mascot Sammy at the franchise location, and so on. Anita reports that 48% of it is customers order their food to go. If this proportion is correct, what is the probability that, in a random sample of 4 customers at Anita’s exactly 2 order their food to go.?

Round your response to at least three decimals places.

7. Suppose that 45% of all babies born in a hospital are boys. If 7 babies born in the hospital are randomly selected, what is the probability that fewer than 2 of them are boys?

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

8. Loretta who turns eighty this year, has just learned about blood pressure problems in the elderly and is interested in how her blood pressure compares to those of her peers. Specifically, she is interested in her systolic blood pressure, which can be problematic among the elderly. She has uncovered an article in a scientific journal that reports that the mean systolic blood pressure measurement for women over seventy -five is 134.3 mmHg, with a standard deviation of 5.6 mmHg.

Assume that the article reported correct information. Complete the follow statement about the distribution of systolic blood pressure measurement for women over seventy-five

a. According to Chebyshev’s theorem, at least (what percentage) of the measurement lie between 125.9 mmHg and 142.7 mmHg.

b. According to Chebyshev’s theorem, at least (what percentage) of the measurement lie between 123.1 mmHg and 145.5 mmHg.

c. Suppose that the distribution is bell shaped. According to the empirical rule, approximately (what percentage) of the measurements lie between 123.1mmHg and 145.5 mmHg

d. Suppose that the distribution is bell-shaped, according to the empirical rule, approximately 99.7% of the measurements lie between ____mmHg and ___mmHg.

9. 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 0, 1, 2, 3, and 4.

Value x of X	P (X = x)
0	0.14
1
2	0.28
3
4	0.21

10. 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 hears, 13 spades, 13 diamonds, and 13 clubs) and draw one card at random. We will ask Clara to name the suite (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 care, 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 suit guesses for each.

Assume that Clare 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 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 decimal places.