8. Populations, Samples, and Probability

Question 1

Population

Answer 1

Any complete set of observations (or potential observations).

Question 2

Sample

Answer 2

Any subset of observations from a population.

Question 3

For each ot the following pairs, indicate with a Yes or No whether the relationship between the first and second expressions could describe that between a sample and its population, respectively.

students in the last row; students in class

Answer 3

Yes, it is a real population.

Question 4

For each ot the following pairs, indicate with a Yes or No whether the relationship between the first and second expressions could describe that between a sample and its population, respectively.

citizens of Wyoming; citizens of New York

Answer 4

No. Citizens of Wyoming aren’t a subset of citizens of New York.

Question 5

20 lab rats in an experiment; all lab rats, similar to those used, that could undergo the same experiment

Answer 5

Yes, it is a hypothetical population.

Question 6

For each ot the following pairs, indicate with a Yes or No whether the relationship between the first and second expressions could describe that between a sample and its population, respectively.

all U.S. presidents; all registered Republicans

Answer 6

No. All U.S. presidents aren’t a subset of all registered Republicans.

Question 7

For each ot the following pairs, indicate with a Yes or No whether the relationship between the first and second expressions could describe that between a sample and its population, respectively.

two tosses of a coin; all possible tosses of a coin

Answer 7

Yes, it is a hypothetical population.

Question 8

Random Sampling

Answer 8

A selection process that guarantees all potential observations in the population have an equal chance of being selected.

Question 9

Indicate whether the following statement is True or False. A random selection of 10 playing cards from a deck of 52 cards implies that

the random sample of 10 cards accurately represents the important features of the whole deck.

Answer 9

False. Sometimes, just by chance, a random sample of 10 cards fails to represent the important features of the whole deck.

Question 10

Indicate whether the following statement is True or False. A random selection of 10 playing cards from a deck of 52 cards implies that

each card in the deck has an equal chance of being selected.

Answer 10

True.

Question 11

Indicate whether the following statement is True or False. A random selection of 10 playing cards from a deck of 52 cards implies that

it is impossible to get 10 cards from the same suit (for example, 10 hearts)

Answer 11

False. Although unlikely, 10 hearts could appear in a random sample of 10 cards.

Question 12

Indicate whether the following statement is True or False. A random selection of 10 playing cards from a deck of 52 cards implies that

any outcome, however unlikely, is possible.

Answer 12

True.

Question 13

Describe how you would use the table of random numbers to take a random sample of five statistics students in a classroom where each of nine rows consists of nine seats.

Answer 13

There are many ways. For instance, consult the tables of random numbers, using the first digit of each 5-digit random number to identify the row (previously labeled
1, 2, 3, and so on), and the second digit of the same random number to locate a particular student’s seat within that row. Repeat this process until five students
have been identified. (If the classroom is larger, use additional digits so that every student can be sampled.)

Question 14

Describe how you would use the table of random numbers to take a random sample of size 40 from a large directory consisting of 3041 pages, with 480 lines per page.

Answer 14

Once again, there are many ways. For instance, use the initial 4 digits of each random number (between 0001 and 3041) to identify the page number of the telephone
directory and the next 3 digits (between 001 and 480) to identify the particular line on that page. Repeat this process, using 7-digit numbers, until 40 telephone
numbers have been identified.

Question 15

Random Assignment

Answer 15

A procedure designed to ensure that each subject has an equal chance of being assigned to any group in an experiment.

Question 16

Assume that 12 subjects arrive, one at a time, to participate in an experiment. Use random numbers to assign these subjects in equal numbers to group A and group B. Specifically, random numbers should be used to identify the first subject as either A or B, the second subject as either A or B, and so forth, until all subjects have been identified. There should be six subjects identified with A and six with B.

Formulate an acceptable rule for single-digit random numbers. Incorporate into this rule a procedure that will ensure equal numbers of subjects in the two groups.

Answer 16

For instance, if the first digit is odd (1, 3, 5, 7, or 9), the first subject is assigned to group A, and if the first digit is even (0, 2, 4, 6, or 8), the first subject is assigned
to group B. To ensure equal groups, the second subject is assigned automatically to the group opposite that for the first subject. Repeat this procedure for the remaining five pairs of subjects.

There are other acceptable rules, all involving pairs of subjects (to ensure equal group sizes). For instance, if the first digit equals 0, 1, 2, 3, or 4, the first subject is assigned to group A; otherwise, the first subject is assigned to group B, and so on.

Question 17

Probability

Answer 17

The proportion or fraction of times that a particular event is likely to occur.

Question 18

Mutually Exclusive Events

Answer 18

Events that cannot occur together.

Question 19

Addition Rule

Answer 19

Add together the separate probabilities of several mutually exclusive events to find the probability that any one of these events will occur.

Question 20

Addition Rule for Mutually Exclusive Events

Answer 20

Pr(A or B) = Pr(A) + Pr(B)

where A and B are mutually exclusive events.

Question 21

Assuming that people are equally likely to be born during any one of the months, what is the probability of Jack being born during…

June?

Answer 21

1/12

Question 22

Assuming that people are equally likely to be born during any one of the months, what is the probability of Jack being born during…

any month other than June?

Answer 22

11/12

Question 23

Assuming that people are equally likely to be born during any one of the months, what is the probability of Jack being born during…

either May or June?

Answer 23

2/12

Question 24

Independent Events

Answer 24

The occurence of one event has no effect on the probability that the other event will occur.

Question 25

Multiplication Rule

Answer 25

Multiply together the separate probabilities of several independent events to find the probability that these events will occur together.

Question 26

Multiplication Rule for Independent Event

Answer 26

Pr(A and B) = [Pr(A)][Pr(B)]

where A and B are independent events.

Question 27

Assuming that people are likely to be born during any of the months, and also assuming (possibly over the objections of astrology fans) that the birthdays of married couples are independent, what’s the probability of…

the husband being born during January and the wife being born during February?

Answer 27

1/12 * 1/12 = 1/144

Question 28

Assuming that people are likely to be born during any of the months, and also assuming (possibly over the objections of astrology fans) that the birthdays of married couples are independent, what’s the probability of…

both husband and wife being born during december?

Answer 28

1/12 * 1/12 = 1/144

Question 29

Assuming that people are likely to be born during any of the months, and also assuming (possibly over the objections of astrology fans) that the birthdays of married couples are independent, what’s the probability of…

both husband and wife being born during the spring (April or May)?

Answer 29

(1/12 + 1/12) * 2/12 = 4/144 or 1/36

Question 30

Conditional Probability

Answer 30

The probability of one event, given the occurence of another event.

Question 31

Among 100 couples who had undergone marital counseling, 60 couples described their relationships as improved, and among this latter group, 45 couples
had children. The remaining couples described their relationships as unimproved, and among this group, 5 couples had children.

What is the probability of randomly selecting a couple who described their relationship as improved?

Answer 31

(Hint: Using a frequency analysis, begin with the 100 couples, first branch into the number of couples with improved and unimproved relationships, then under each of these numbers, branch into the number of couples with children and without children. Enter a number at each point of the diagram before proceeding.)

60/100 = 0.60

Question 32

Among 100 couples who had undergone marital counseling, 60 couples described their relationships as improved, and among this latter group, 45 couples had children. The remaining couples described their relationships as unimproved, and among this group, 5 couples had children.

What is the probability of randomly selecting a couple with children?

Answer 32

(45 + 5) / 100 = .50

Question 33

Among 100 couples who had undergone marital counseling, 60 couples described their relationships as improved, and among this latter group, 45 couples had children. The remaining couples described their relationships as unimproved, and among this group, 5 couples had children.

What is the conditional probability of randomly selecting a couple with children, given that their relationship was described as improved?

Answer 33

45/60 = .75

Question 34

Among 100 couples who had undergone marital counseling, 60 couples described their relationships as improved, and among this latter group, 45 couples had children. The remaining couples described their relationships as unimproved, and among this group, 5 couples had children.

What is the conditional probability of randomly selecting a couple without children, given that their relationship was described as not improved?

Answer 34

35/40 = .875

Question 35

Among 100 couples who had undergone marital counseling, 60 couples described their relationships as improved, and among this latter group, 45 couples had children. The remaining couples described their relationships as unimproved, and among this group, 5 couples had children.

What is the conditional probability of an improved relationship, given that a couple has children?

Answer 35

45 / (45 + 5) = 0.90