If a and B Are Mutually Exclusive Events with P(A) = 0.70, Then P(B)

If A and B are mutually exclusive events with P(A) = 0.70, then P(B)

can be any value between 0 and 1.

can be any value between 0 and 0.70.

cannot be larger than 0.30.

None of the above statements are true.

If A and B are independent events with P(A) = 0.20 and P(B) = 0.60, then P(A|B) is

0.2000

In the notation below, X is the random variable, c is a constant, and V refers to the variance. Which of the following laws of variance is not correct?

V(c) = 0

V(X + c) = V(X)

V(X + c) = V(X) + c

V(cX) = c2 V(X)

Which of the following statements is always correct?

P(A and B) = P(A) * P(B)

P(A or B) = P(A) + P(B)

P(A or B) = P(A) + P(B) + P(A and B)

P= 1- P(A)

An experiment consists of tossing an unbiased coin three times. Drawing a probability tree for this experiment will show that the number of simple events in this experiment is

Use the following information to answer questions 6 and 7: The weights of newborn children in the United States vary according to a normal distribution with mean 7.5 pounds and standard deviation 1.25 pounds. The government classifies a newborn as having low birth weight if the weight is less than 5.5 pounds.

What proportion of babies weigh less than 5.5 pounds at birth?

0.0548

If the government wanted to change the value 5.5 pounds to a weight where only 2% of newborns weigh less than the new value, what weight should they use?

4.9375 pounds

If you are given a table of joint probabilities of two events, any probability computed by adding across rows or down columns is also called

marginal probability

joint probability

conditional probability

Bayes’ theorem

The effect of increasing the standard deviation of a normally distributed random variable is that the distribution becomes

narrower and more peaked.

flatter and wider.

If the random variable X follows a Uniform distribution with a=7.5 and b=11.5, what is P(X<10)?

0.625

The weighted average of the possible values that a discrete random variable X can assume, where the weights are the probabilities of occurrence of those values, is referred to as the

variance.

standard deviation.

expected value.

covariance.

The variance of a binomial distribution for which n = 100 and p = 0.20 is

16.

A professor receives, on average, 28.6 e-mails from students the day before the final exam. To compute the probability of receiving at least 8 e-mails on such a day, he will use what type of probability distribution?

Binomial distribution

Poisson distribution

Normal distribution

Exponential distribution

Given that X is a normally distributed random variable, which of the following statements is true?

The variable X + 5 is also normally distributed.

The variable X - 5 is also normally distributed.

The variable 5X is also normally distributed.

All of the above.

If two events are mutually exclusive, what is the probability that both occur at the same time?

0.00

The chancellor of a major university was concerned about alcohol abuse on her campus and wanted to find out the percentage of students at her university who visited city bars every weekend. Her advisor took a random sample of 250 students. The percentage of students in the sample who visited city bars every weekend is an example of

a categorical random variable.

a discrete random variable.

a continuous random variable.

a parameter.

If X and Y are any random variables, which of the following identities is not always true?

E (X+Y) = E(X) + E(Y)

V(X+Y) = V(X) + V(Y)

E(4X+5Y) = 4E(X) + 5 E(Y)

V(4X+5Y) = 16V(X) + 25V(Y) + 40COV(X,Y)

If the random variable X follows an exponential distribution with , what is P(X10)?

0.3935

The joint probabilities shown in a table with two rows, and , and two columns, and , are as follows: P( and ) = .10, P( and ) = .30, P( and ) = .05, and P(and ) = .55. Then P()

0.40.

If the random variable X follows a Poisson distribution with, what is P(X5)?

0.868

An effective and simple method of applying the basic probability rules is the

probability tree.

Suppose Z is a random variable with a standard normal distribution.

What proportion of observations from Z are greater than -0.13?

z-0.13 = 0.4483

P(z>-0.13) = 1-0.4483 = 0.5517

What is P(0.29Z1.45)?

z1.45 = 0.9265, z0.29 = 0.6141

P(0.29£Z£1.45) = 0.9265-0.6141 = 0.3164

What is the twentieth percentile of Z? [In other words, find the value of z such that approximately 20% of standard normal observations fall below it.]

P(Z£ -0.84)

What is the third quartile of Z? [In other words, find the value of z such that the proportion of standard normal observations falling below z is approximately 75%.]

0.67

In a shipment of 15 room air conditioners, there are 4 with defective thermostats. Two air conditioners will be selected at random (without replacement) and inspected one after the other. Find the probability that: (Hint: A probability tree may help.)

The first is defective.

P(1st defective = 4/15 = 0.267)

The first is defective and the second one is good.

P(1st D and 2nd G) = 0.79 * 0.27 = 0.2133

Both are defective.

P( 1st D and 2nd D) = 0.27 * 0.21 = 0.0567

The second one is defective, given that the first one was good.

P( 2nd D | 1st G) = 4/14 = 0.2857

Exactly one is defective.

P(exactly one D) = P( 1st D or 2nd D) = P(1st D) + P(2nd D) = 0.2667 + 0.2667 = 0.5334

QUIZZES CH6- Which of the following statements is always correct?

A. P(A and B) = P(A)P(B)

B. P(A or B) = P(A) + P(B)

C. P(A or B) = P(A) + P(B) + P(A and B)

D. P (Ac) = 1-P(A)

If a coin is tossed three times and a statistician predicts that the probability of obtaining three heads in a row is 0.125, which of the following assumptions is irrelevant to his prediction?

A. The events are dependent.

B. The events are independent.

C. The coin is unbiased.

D. All of these choices.

The manager of the customer service division of a major consumer electronics company is interested in determining whether the customers who have purchased a videocassette recorder made by the company over the past 12 months are satisfied with their products. The possible responses to the question "How many videocassette recorders made by other manufacturers have you used"? are values from a:

discrete random variable.

continuous random variable.

categorical random variable.

parameter.

If X and Y are any random variables with E(X) = 50, E(Y) = 6, E(XY) = 21, V(X) = 9 and V(Y) = 10, then the relationship between X and Y is a:

strong positive relationship.

strong negative relationship.

weak positive relationship.

weak negative relationship.

categorical random variable.

continuous random variable.

parameter.

discrete random variable

If X and Y are random variables with E(X) = 5 and E(Y) = 8, then E(2X + 3Y) is:

A function or rule that assigns a numerical value to each simple event of an experiment is called:

a sample space.

a probability tree.

a probability distribution.

a random variable

If X is a binomial random variable with n = 25, and p = 0.25, then the variance and standard deviation of X are 6.25 and 2.5, respectively.

True

False

The number of customers arriving at a department store in a 5-minute period has a Poisson distribution.

True

False

CH6 REC WS- Assume that the weights of bags of potato chips are independent of one another. Assume that the third quartile is 19 oz. Suppose I randomly select 4 bags of chips. P(W>18)=1/4, P(W<18)=3/4

What is the probability that all four bags weigh more than 18 oz?

P(All 4 bags > 18) = P(1st > 18) P(2st > 18) P(3st > 18) P(4st > 18)

What is the probability that none of the 4 bags weigh more than 18 oz?

P(NONE>18) + P(ALL < 18) = P(1st < 18) P(2st < 18) P(3st < 18) P(4st < 18)

What is the probability that at least one of the 4 bags weighs more than 18 oz?

P(At least 1 bag > 18) = 1 – P(NONE>18) = 1 – (¾)4

CH6

1) A random experiment is an action or process that leads to one of several possible outcomes

2) A sample space of a random experiment is a list of all possible outcome of the experiment. The outcomes must me exhaustive (all possible outcomes must be included) and mutually exclusive (no two outcomes can occur at the same time).

3) Requirements of probabilities: Given a sample space S={O1,O2,…,Ok}, the probabilities assigned to the outcomes must satisfy two requirements:

a. The probability of any outcome must lie between 0 and one 0P(Oi)1 for each i

b. The sum of the probabilities of all the outcomes in a sample space must be one.

4) Classical approach is associated with games of chance; relative frequency approach defines probabilities as the long-run frequency with which an outcome occurs; subject approach defines probability as the degree of belief that we hold in the occurrence of an event

5) An individual outcome of a sample space is called a simple event; an event is a collection or set of one or more simple events in a sample space

6) The probability of an event is the sum of probabilities of the simple events that constitute the event

1) Intersection of events A and B- the intersection is the event that occurs when both A and B occur (denoted as: A and B); the prob. Of the int. is called the joint probability

2) Conditional probability- the probability of event A given event B is P(A|B)=P(A and B)/P(B)

3) Independent events- two events A and B are said to be independent if P(A|B)=P(A)

4) The union of events A and B is the event that occurs when either A or B or both occur.

1) Complement Rule- P(AC)=1-P(A)

2) The multiplication rule is used to calculate the joint probability of two events. P(A and B)= P(A|B)P(B)

a. The joint probability of any two independent events A and B is P(A and B)=P(A)P(B)

3) The addition rule calculate the probability of the union of two events P(A or B)=P(A)+P(B)-P(A and B)

a. The probability of the union of two mutually exclusive events A and B is
P(A or B)=P(A)+P(B)

CH7

1) A random variable is a function or rule that assigns a number to each outcome of an experiment often labeled as (X). The number of heads when a coin is tossed is the random variable or value of X.

a) Discrete random variable- one that can take on a countable number of values

i) 0P(x)1 for all x

ii) e (of all x) P(x)=1

b) Continuous random variable0 values are uncountable

i) Probability distribution- table, formula, or graph that describes the values of a RV and the probability associated with these values

2) Statistical inference deals with inferences about populations

a) The population mean (expected value) is the weighted average of all its values, the weights are the probabilities E(X)=µ= e xP(x)

i) =0P(0)+1P(1)+2P(2)+…. nP(n)

b) Population variance is the weighted average of the squared deviations from the mean
V(X)= δ2= e (x-µ) 2P(x) or V(X)= δ2= e x 2P(x)-µ2

i) =(0-µ) 2 P(0)+ (1-µ) 2 P(1)+ … (n-µ) 2 P(n) or = 02 P(0)+ 12 P(1) … n2 P(n) - µ2

c) Standard Deviation is δ=square root of δ2

3) Laws of Expected Value Laws of Variance

a) E(c)=c V(c)=0

b) E(X+c)=E(X)+c V(X+c)=V(X)

c) E(cX)=cE(X) V(cX)=c2V(X)

1) Bivariate distribution- probabilities combinations of two variables. The joint probability that the two variables will assume the values x and y is denoted P(x, y).

a. A bivariate or joint probability distribution of X and Y is a table or formula that lists the joint probabilities for all pairs of values of X and Y.

b. Requirements for discrete bivariate distribution: 0P(x, y) 1 for all pairs of values (x,y) and e (all x) e (all y) P(x,y)=1

i. Covariance

ii. Coefficient of Correlation

c. Sum of 2 variables P(X+Y=2)=P(0,2)+P(1,1)+P(2,0)=.07+.06+.06=.19
x+y | 0 1 2 3 4
P(x+y)| .12 .63 .19 .05 .01

d. Laws of expected value Variance of the sum of two variables

E(X+Y)=E(X)+E(Y) V(X+Y)=V(X)=V(Y)+2COV(X,Y)
*If x and y are ind., COV (X,Y)=0 and thus V(X+)=V(X)+V(Y)

2) Binomial experiment - discrete

a. Consists of a fixed number of trials. We represent the number of trials by n.

b. On each trial there are two possible outcomes. We label one outcome as a success, and the other as a failure

c. The probability of success is p. The probability of failure is 1-p.

d. The trials are independent, which means that the outcome of one trial does not affect the outcome of any other trial.

i. The random variable of a binomial experiment is defined as the number of successes in the n trials. It is called the binomial random variable.

ii. Must REPLACE the card to be binomial- if you don’t they’re not independ.

e. The probability of x successes in a binomial experiment with n trials and probability of success = p is

P(x)=……………….

f. Finding the binomial probability P(Xx) = 1-P(X[x-1])

g. Finding the probability P(X=x) = P(Xx)-P(X<[x-1])