Extending the variance of the sum rule. For mathematical convenience we first extend the sum rule to three random variables with zero

expectation. Next we further extend the rule to three random variables with

nonzero expectation. By the same line of reasoning we extend the rule to n

random variables.

a. Let X, Y and Z be random variables with expectation 0. Show that

Var(X + Y + Z) = Var(X) + Var(Y ) + Var(Z)

+ 2Cov(X, Y ) + 2Cov(X, Z) + 2Cov(Y,Z).

Hint: directly apply that for real numbers y1,...,yn

(y1 + + yn)

2 = y2

1 + + y2

n + 2y1y2 + 2y1y3 + + 2yn?1yn.

b. Now show a for X, Y , and Z with nonzero expectation.

Hint: you might use the rules on pages 98 and 141 about variance and

covariance under a change of units.

c. Derive a general variance of the sum rule, i.e., show that if X1, X2,...,Xn

are random variables, then

Var(X1 + X2 + + Xn)

= Var(X1) + +Var(Xn)

+2Cov(X1, X2) + 2Cov(X1, X3) + + 2Cov(X1, Xn)

+ 2Cov(X2, X3) + + 2Cov(X2, Xn)

...

+ 2Cov(Xn?1, Xn).

d. Show that if the variances are all equal to ?2 and the covariances are all

equal to some constant ?, then

Var(X1 + X2 + + Xn) = n?2 + n(n ? 1)?.

10.18 Consider a vase containing balls numbered 1, 2,...,N. We draw

n balls without replacement from the vase. Each ball is selected with equal

probability, i.e., in the first draw each ball has probability 1/N, in the second

draw each of the N ? 1 remaining balls has probability 1/(N ? 1), and so

on. For i = 1, 2,...,n, let Xi denote the number on the ball in the ith draw.

From Exercise 9.18 we know that the variance of Xi equals

Var(Xi) = 1

12(N ? 1)(N + 1).150 10 Covariance and correlation

Show that

Cov(X1, X2) = ? 1

12(N + 1).

Before you do the exercise: why do you think the covariance is negative?

Hint: use Var(X1 + X2 + + XN ) = 0 (why?), and apply Exercise 10.17.

10.19 Derive the alternative expression for the covariance: Cov(X, Y ) =

E[XY ] ? E[X]E[Y ].

Hint: work out (X ? E[X])(Y ? E[Y ]) and use linearity of expectations.

10.20 Determine ?

U, U2

when U has a U(0, a) distribution. Here a is a

positive number.

Rank the scales of measurement in order from

least informative to most informative.

23. What is the main disadvantage of measuring qualitative data? In your answer, explain why quantitative research is most often applied in the

behavioral sciences.

24. State whether each of the following describes a

study measuring qualitative or quantitative

data.

a. A researcher distributed open-ended questions

to participants asking how they feel when

they are in love.

b. A researcher records the blood pressure of

participants during a task meant to induce

stress.

c. A psychologist interested in drug addiction

injects rats with an attention-inducing drug

and then measures the rate of lever pressing.

d. A witness to a crime gives a description of the

suspect to the police.

25. State whether each of the following are continuous or discrete data.

a. Time in seconds to memorize a list of words

b. Number of students in a statistics class

c. The weight in pounds of newborn infants

d. The SAT score among college students

26. Fill in the table below to identify the characteristics of each variable.

7. About 60 % of people living in a country wear glasses. When you pick 8 persons randomly from this population, what is the the probability that (a) exactly 2 wear glasses? [Show manual computation.) (b) 7 or more wear glasses. [Show manual computation.] (c) Let X be the number of those who wear glasses among the 8 people selected. What is the range of X? Is the distribution skewed or symmetric? If it is skewed, in what direction? (d) Find the mean and standard deviation of the random variable X. [The answers require very little or no computation.]8. Gro-More Pet Food has a 16% market share. Assume the probability a customer will buy Gro- More is 16%. A sample of 24 customers is selected. a) Consider a binomial random variable X indicating the number of customers in a sample of 24 who will buy Gro-More. What values can the random variable assume? b) Find the probability that exactly eight customers in the sample of 24 will buy Gro-More. Express your answer to four places after the decimal point. c) Find the probability that at least three customers out of the sample of 24 will buy Gro-More. Express your answer to four places after the decimal point. d) Find the expected value of the random variable X. e) Find the standard deviation of the random variable X.21. Suppose that the miles-per-gallon (mpg) rating of passenger cars is a normally distributed random variable with a mean and a standard deviation of 33.8 and 3.5 mpg, respectively. Use Table 1. a. What is the probability that a randomly selected passenger car gets more than 35 mpg? (Round lnterrnedlate calculations to 4 decimal places, "2" value to 2 decimal places, and final answer to 4 decimal places.) Probability |:| b. What is the probability that the average mpg of four randomly selected passenger cars is more than 35 mpg? (Round intermediate calculations to 4 decimal places, \"2\" value to 2 decimal places, and nal answer to 4 decimal places.) Probability U c. If four passenger cars are randomly selected. what is the probability that all of the passenger cars get more than 35 mpg? (Round intermediate calculations to 4 decimal places, "2\" value to 2 decimal places, and nal answer to 4 decimal places.) Probability |:|