ch. 9 statistics t.h. q
The Wechsler Adult Intelligence Scale (WAIS) is a standardized test of intellectual ability, arguably measuring intelligence. Based on normative data collected from the general population, it has been determined to have a mean of 100 and a standard deviation of 15. The probability of a person selected at random scoring >115 (one S.D. above the mean) is, according to the empirical rule, approximately .16 (check this, if you're not sure how that is determined, by reviewing the empirical rule). If we let this represent event B, then P(B) = .16. P(A and B) = P(A) x P(B) = __....
. 10 x .16 = .016
A teaching assistant with a lecture class of 100 students must select one student to act as moderator for a discussion section. She would prefer to have either a sophomore or junior student for this role. If her class has 40 sophomores and 30 juniors and she makes a random selection from the class roster, the probability that the student selected is either a sophomore or junior can be determined as follows: P(sophomore or junior) = P(A) + P(B) = 4/10 P(B) = P (English major) =
. 40 + .30 = .70
The probability that a psychology major will take Introduction to Statistics is approximately .9 (since most psychology programs require a course in statistics). Of students who have completed an introductory statistics course, approximately 25% will go on to take a more advanced course, Research Methods. Introduction to Statistics is a prerequisite for Research Methods. Step 1: P(A) = .9, the probability that a psychology major will take Introduction to Statistics. Step 2: Step 2: P(B | A) = ___, the probability that a psychology major who has completed Introduction to Statistics will go on to take Research Methods.
.25
An online survey has boosted participation by offering a prize to every 20th participant who completes a survey. How would the likelihood of receiving a prize be stated as odds? In the previous question, what is the probability of receiving a prize?
.5
In a pet store there are 6 puppies, 9 kittens, 4 gerbils, and 7 parakeets. If a pet is chosen at random, what is the probability of choosing a puppy or a parakeet?
.5
The probability of a teenager in Toronto owning a skateboard is .37, of owning a bicycle is .81, and of owning both is .36. If a Toronto teenager is chosen at random, what is the probability that the teenager owns a skateboard or a bicycle?
.82
85% of the students at a certain community college have graduated. 35% of those graduates go on to complete bachelor's degrees. What is the probability that a student drawn at random will complete a bachelor's degree?
0.2975
A "sure thing" would have a probability equal to _____ .
1.0
According to data from the American Cancer Society, about 1 in 3 women living in the U.S. will have some form of cancer during their lives. If three women are randomly selected, what is the probability that they will all contract cancer at some point during their lives?
1/27
P(A) = Rolling a six
1/6
P(A and B) = P(A) x P(B) =
1/6 x 1/4 = 1/24
The access code for a security door consists of four digits. Each digit can be any number from 0 through 9. How many access codes are possible?
10000.0
P(B) = drawing a spade =
13/52 = 1/4
An online survey has boosted participation by offering a prize to every 20th participant who completes a survey. How would the likelihood of receiving a prize be stated as odds?
1:19
Identify the true statement. A. Events that are not mutually exclusive are also not complementary B. Events with shared elements are mutually exclusive C. Mutually exclusive events are also not complementary D. Dependent events are also not mutually exclusive
A.
In a group of students, 40 are juniors, 50 are English majors,and 22 are English-major juniors. The 22 English juniors represent elements shared by the two categories "juniors" and "English majors" and could be represented graphically as the intersection of those categories.
Non-Exclusive
(Multiplication Rule) If events are independent, the probability that they will all occur is equal to the product of their individual probabilities. (The product is the result of multiplying two or more numbers.) This is the most basic application of the multiplication rule, and can be stated as:
P(A and B) = P(A) x P(B)
These are non examples of.... - "A football player *runs for a touchdown* " is not a probability experiment. - "A person *buys a raffle ticket* " is not a probability experiment. - "Your church sponsors a Las Vegas night to *raise money for charity*" is not a probability experiment.
Probability Experiment
This is an ex. of what kind of probability? For example, "the probability of *winning the lottery* is .5" may not accurately reflect the *reality of the situation.*
Subjective
Probability is key to assessing _______
risks.`
This is an ex. of which probability? For example, the *probability of rolling a two on a six-sided die is one sixth*
theoretical
based on an *assumption about the nature of the event* , in which it is assumed that n events are equally likely to occur.
theoretical (classical) probability
There are ten marbles—three blue, three red, and four green—in a bag. To find the probability of randomly selecting a blue marble after a green has already been selected and removed from the bag, we could use the formula for conditional probability, since these are dependent events (the probability of selecting blue changes as a result of the green having been removed). Step 1: P(A) = selecting a green marble = 4/10 Step 2: P(B | A) = selecting a blue marble after selecting a green = ___, ___
3/9 = 1/3
A teaching assistant with a lecture class of 100 students must select one student to act as moderator for a discussion section. She would prefer to have either a sophomore or junior student for this role. If her class has 40 sophomores and 30 juniors and she makes a random selection from the class roster, the probability that the student selected is either a sophomore or junior can be determined as follows: P(B) = P(junior) =
30/100 = .30
Can be used to find the probability that at least one of two events will occur. Note how this differs from the multiplication rule: the multiplication rule is used to find the probability that both (or all) events will occur, while the ___________ ____ is used to find the probability the one or the other of two events will occur.
Addition Rule
E,S, or T? *The likelihood* of a coin coming up heads
Theoretical
Two __________ events, when combined, make up the entire sample space.
complementary
The probability of event B given that event A has occurred is known as ________ probability.
conditional
Two events are _________ __________ *if they cannot occur at the same time.* Note that this *does not imply they are independent events*, only that they cannot occur simultaneously. (same definition) events could be either dependent or independent.
mutually exclusive
Pumping gas into your car and driving is a (mutually exclusive/non-exclsuive), (independent/dependent) event.
mutually exclusive, dependent
Cramming for an exam and training for a bicycle race, is a (mutually exclusive/non-exclsuive), (independent/dependent) event.
mutually exclusive, independent
Two events are __________ _______ if they can occur at the same time. One event could be a subset of the other (complete overlap), or the events may have some proportion of shared elements. They have shared elements that are common to both A and B.
nonexclusive events
the description of *all possible outcomes* of a probability experiment. (total outcomes)
sample space
In a probability experiment involving the roll of a die, the event "rolling a number greater than five" is a ________ event.
simple
- A _______ event consists of a *single outcome.* - A ___________ event consists of *more than one outcome.*
simple, compound
The probability that a psychology major will take Introduction to Statistics is approximately .9 (since most psychology programs require a course in statistics). Of students who have completed an introductory statistics course, approximately 25% will go on to take a more advanced course, Research Methods. Introduction to Statistics is a prerequisite for Research Methods. Step 1: P(A) = __, the probability that a psychology major will take Introduction to Statistics.
.9
A day of the week is chosen at random. What is the probability of choosing a weekend day?
2/7
In a previous example we showed a group of students, of whom 40 are juniors, 50 are English majors, and 22 are English-major juniors. The 22 English juniors represent elements shared by the two categories "juniors" and "English majors" and could be represented graphically as the intersection of those categories. Since the categories have shared elements they are not mutually exclusive. A student can be both an English major and a junior. If there were 100 students total and we wanted to find the probability of selecting either a junior or a Psychology major student, we could proceed as follows: P(sophomore or junior) = P(A) + P(B) = 4/10 P(B) = P (English major) = 50/100 P(A) = P(junior) = __/____
22/100
In the sample space for the probability experiment of rolling a pair of dice, how many outcomes are there?
36. (6 * 6)
(exe.) The probability experiment of *rolling a single die* to see if a *number greater than 2 comes up has how many outcomes*?
4.0
There are ten marbles—three blue, three red, and four green—in a bag. To find the probability of randomly selecting a blue marble after a green has already been selected and removed from the bag, we could use the formula for conditional probability, since these are dependent events (the probability of selecting blue changes as a result of the green having been removed). Step 1: P(A) = selecting a green marble = ????
4/10
There are ten marbles—three blue, three red, and four green—in a bag. To find the probability of randomly selecting a blue marble after a green has already been selected and removed from the bag, we could use the formula for conditional probability, since these are dependent events (the probability of selecting blue changes as a result of the green having been removed). Step 1: P(A) = selecting a green marble = 4/10 Step 2: P(B | A) = selecting a blue marble after selecting a green = 3/9 = 1/3 Step 3: P(A and B) = P(A) × P(B | A) = ___.....
4/10 × 1/3 = 4/30 = 2/15
In a previous example we showed a group of students, of whom 40 are juniors, 50 are English majors, and 22 are English-major juniors. The 22 English juniors represent elements shared by the two categories "juniors" and "English majors" and could be represented graphically as the intersection of those categories. Since the categories have shared elements they are not mutually exclusive. A student can be both an English major and a junior. If there were 100 students total and we wanted to find the probability of selecting either a junior or a Psychology major student, we could proceed as follows: P(sophomore or junior) = P(A) + P(B) = _/___
40/100
In a previous example we showed a group of students, of whom 40 are juniors, 50 are English majors, and 22 are English-major juniors. The 22 English juniors represent elements shared by the two categories "juniors" and "English majors" and could be represented graphically as the intersection of those categories. Since the categories have shared elements they are not mutually exclusive. A student can be both an English major and a junior. If there were 100 students total and we wanted to find the probability of selecting either a junior or a Psychology major student, we could proceed as follows: P(sophomore or junior) = P(A) + P(B) = 4/10 P(B) = P (English major) = 50/100 P(A) = P(junior) = 22/100 P(A or B) = P(A) + P(B) - P(A and B) =
40/100 + 50/100 - 22/100 =68/100 or .68
A teaching assistant with a lecture class of 100 students must select one student to act as moderator for a discussion section. She would prefer to have either a sophomore or junior student for this role. If her class has 40 sophomores and 30 juniors and she makes a random selection from the class roster, the probability that the student selected is either a sophomore or junior can be determined as follows: P(A) = P(sophomore) =
40/100 = .40
In a previous example we showed a group of students, of whom 40 are juniors, 50 are English majors, and 22 are English-major juniors. The 22 English juniors represent elements shared by the two categories "juniors" and "English majors" and could be represented graphically as the intersection of those categories. Since the categories have shared elements they are not mutually exclusive. A student can be both an English major and a junior. If there were 100 students total and we wanted to find the probability of selecting either a junior or a Psychology major student, we could proceed as follows: P(B) = P (English major) = __/____
50/100
The probability that a psychology major will take Introduction to Statistics is approximately .9 (since most psychology programs require a course in statistics). Of students who have completed an introductory statistics course, approximately 25% will go on to take a more advanced course, Research Methods. Introduction to Statistics is a prerequisite for Research Methods. Step 1: P(A) = .9, the probability that a psychology major will take Introduction to Statistics. Step 2: Step 2: P(B | A) = .25, the probability that a psychology major who has completed Introduction to Statistics will go on to take Research Methods. Step 3: P(A and B) = P(A) x P(B | A), ___ _ ___ = ___ , the probability that a psychology major will take Research Methods.
= .9 x .25 = .225
Which of the following is *not* a probability experiment? A. Wagering on the outcome of a spin of a roulette wheel B. Guessing the answer to a multiple choice quiz question C. A soccer player attempting a goal D. Selecting a square on the office football pool
C. A soccer player attempting a goal
One marble is drawn from a jar containing 4 red, 6 blue, and 7 green marbles. What is the sample space? A. {4, 6, 7} B. 4, 6, 7 C. {red, blue, green} D. {4 red, 6 blue, 7 green}
C. {red, blue, green}
All *outcomes that are not the event* .
Complements of Events
Which value below could *not* be a probability? A. .54 B. 0 C. .12 D. -.12
D. -12.
These are events of what kind? - If a queen is drawn from a deck of cards on the first trial *and is not replaced, the probability of drawing another queen on the second trial is changed as a result of the outcome of the first trial.* Prior to the first trial there were four queens in the deck, but after one was removed there were only three available for the second trial. The probability of drawing a queen changed from 4/52 on the first trial to 3/51 on the second trial. - Health psychologists have related behavioral risk factors to health-related outcomes. One such is excessive consumption of alcohol, which has been found to be related to an increased incidence of liver cancer. Alcohol consumption and the development of liver cancer are, in that respect, ___________ events, since *the probability of developing liver cancer is affected (changed) by the occurrence of excessive alcohol consumption.* Note that this is not to say that liver cancer will develop as a result of excessive alcohol consumption, only that the probability is affected by the prior behavior.
Dependent
What event would we define this as? Cigarette smoking and developing lung cancer
Dependent
the probability that one will occur *is affected (changed) by the occurrence of the other* .
Dependent Events
E,S or T? die is *rolled ten times and the number of threes recorded*
Empirical
This is an ex. of what kind of probability? For example, a die is rolled 10 times and the number two comes up two times, making the relative frequency 2/10.
Empirical
Based on the *observed outcomes of one or a series* of trials. It is the type *used most often in statistical inference procedures*
Empirical (Relative Frequency) Probability
What probability would describe this scenario? A school psychologist is interested in the number of children that families in a certain town have enrolled in elementary school. She sends out 1000 surveys, of which 800 are returned. Of those 800, the data reported are: (data) *Based on these data, the psychologist could conclude* that the probability of a family in that town having two children enrolled in elementary school is P(E) = P(2) = 350/800 = .4375.
Empirical Probability
A formal definition of an event in statistics is a subset of..... also can consist of one or more outcomes that satisfy a given condition, and can be either simple or compound.
Event
Odds represent a ratio of to non-events.
Events
These are examples of... The event, rolling a four, has *only one possible* outcome: {4}. - It is a simple event. The event, rolling a number that is not a four, has five possible outcomes: {1, 2, 3, 5, 6}. It is a *compound event.*
Events at the Neuromuscular Junction
What event would we define this as? Rolling "snake eyes" after rolling a string of sevens
Independent
the probability that one will occur is *not affected by whether or not the other has occurred* .
Independent Event
These are events of what kind? If a die is rolled twice (two trials) and the numbers recorded, *the outcome of the second trial is not affected by the outcome of the first trial.* The probability of any number from one to six coming up on either trial is one sixth. It is unlikely that the probability of a student registering to take Introductory Psychology and the probability of having Cheerios for breakfast are related in any meaningful way, so it is probably safe to assume they are independent events.
Independent Events
can be used to find the probability that two (or more) events occur in sequence, and takes the fundamental counting principle a step further by taking into account both independent and dependent events "and"
Multiplication rule
These are events of what kind? - Selecting a queen from a card deck, *not replacing it*, and then selecting a king
Mutually Exclusive, Dependent Events
The events getting an A in statistics and getting a C in statistics are (mutually/non exclusive, dependent/independent.)
Mutually Exclusive, Independent
These are events of what kind? Being a dog or a cat.
Mutually Exclusive, Independent (A pet cannot be both a dog and a cat at the same time.)
In another group of students, there are 30 juniors and 35 seniors. At least in terms of class ranking, there are no shared elements, so these categories are
Mutually Exclusive. (A student cannot be both a junior and a senior.)
These are examples of a.... - A die is rolled once *to see if* the number four comes up (probability 1 out of 6, or approximately .167). - A coin is tossed *to see if* a heads occurs (probability 1 out of 2, or .5). - A coin is tossed four times *to measure* how many heads come up (probability 1 out of 2 to the 4th power, or (.5)4).
Probability Experiment
These are the outcomes of a.... - The experiment *must have more than one* possible outcome - Each possible outcome can be *specified in advanced* - The outcome of the experiment is *due to chance*
Probability Experiment
a situation involving chance or probability that *leads to observable and measureable results* called outcomes.
Probability experiment
Denoted by a capital letter, e.g., A, B, or C. The objects or values that comprise the set are *enclosed in braces*
Set
These are examples of... {1, 2, 3, 4, 5, 6}
Set
Based on an *individual's personal belief* about the likelihood of an event occurring.
Subjective
E,S, or T? The Yankees *chances of winning* the series
Subjective
What probability type matches this example? Lotteries have been around for ages, but have more recently become highly sophisticated vehicles for raising funds for a variety of causes and agencies. Alice walks into a convenience store to purchase a lottery ticket. *She thinks, in a rather offhand way, that her chance of winning one of the prizes might be as low as one in a hundred but is willing to invest a couple of dollars anyway, since it is just "loose change."*
Subjective Probability
What probability would describe this scenario? A graduate student is setting up an experiment using laboratory rats to investigate the effect of rewards on learning. In one part of the experiment, the rats are released into a maze with four corridors, one of which leads to a bar-press mechanism where the rat can obtain a food pellet. *Assuming that on the first trial there is an equal likelihood of choosing any of the four corridors* , what is the probability the rat will choose the corridor *leading to the food reward (labeled as corridor D)?*
Theoretical
Name the types of probability
Theoretical Empirical Subjective
*single performance* of a probability experiment.
Trial
These are examples of.... - *In the* die rolling experiment, one roll of the die represents one trial. - *In the* first coin toss experiment, the the coin toss represents one trial. - *In the* second coin toss experiment, the series of four tosses represents one trial.
Trials
In a probability experiment, the set of all possible outcomes *represents the sample space.* A *subset* of those outcomes represents a(n) __________.
event
a mathematical rule that enables us to find the number of ways a combination or series of independent events can occur.
fundamental counting principle
the *result of a single trial* in a probability experiment
outcome
these are examples of.... - In the die rolling experiment, the result of that roll of the die *represents* the outcome. There are six possible outcomes, 1 through 6. - In the first coin toss experiment, the result of that toss *represents* the outcome. There are two possible outcomes, H or T. - In the second coin toss experiment, the result of one series of tosses *represents* the outcome of that trial. There are 16 possible outcomes, shown in the examples below.
outcome
