ST 351 - Statistical Methods (Learning Paths 14-18)

A deck of playing cards has 52 cards, divided into four suits (diamonds, spades, clubs, and hearts). There are an equal number of cards in each suit in this deck. If three cards are drawn, what is the probability that the first two cards drawn are clubs and the last is red? There are three events in this problem: Let event A = first card drawn is a club Let event B = second card drawn is a club C = third card drawn is red. We want to find P(A and B and C) (i.e. the probability that the first card drawn is a club AND the second card drawn is a club AND the third card drawn is a red card). Calculate P(A and B and C) if the cards are drawn without replacement. Write your answer as a decimal rounded to FOUR decimal places.

.0306 - P(A) = 13/52, P(B) = 12/51 (given event A occurred), P(C) = 26/50 (given events A and B both occurred) Applying the general multiplication rule, P(A and B and C) = (13/52)(12/51)(26/50)

A deck of playing cards has 52 cards, divided into four suits (diamonds, spades, clubs, and hearts). There are an equal number of cards in each suit in this deck. If three cards are drawn, what is the probability that the first two cards drawn are clubs and the last is red? There are three events in this problem: Let event A = first card drawn is a club Let event B = second card drawn is a club C = third card drawn is red. We want to find P(A and B and C) (i.e. the probability that the first card drawn is a club AND the second card drawn is a club AND the third card drawn is a red card). Suppose the three events are independent. Use the Multiplication Rule to find P(A and B and C). Round answer to FOUR decimal places.

.25 was*.25*.5 = .0313

An unbalanced coin is tossed two times. On this coin, the probability of getting a heads to land face up on any one toss is 0.4. There are four possible outcomes in the sample space (HH, HT, TH, TT), where HT (for example) means heads on first toss AND tails on second toss. If the outcomes in the sample space are disjoint, use the addition rule to find the probability of getting at least one head on the two tosses. Report your answer to TWO decimal places.

.64 = P(HT) + P(TH) + P(HH) = 0.24 + 0.24 + 0.16

If an outcome can never occur, P(outcome) equals?

0

The probability of an outcome is always between:

0 and 1, inclusive • That is, 0 ≤ P (outcome) ≤ 1

A probability closer to ___ indicates a higher likelihood of occurring.

1

If an outcome will definitely occur, what is P(outcome)?

1

The sum of the probabilities of all possible outcomes in the sample space is?

1

A family is planning on having three children. Is the family more likely to have all of one sex or two of one sex and one of the other? (We'll assume that it is equally likely that a child born is a girl or boy.) 1. What is the probability that all three children are boys? 2. What is the probability that the first born is a boy and the next two born are girls?

1 .125 = .5*.5*.5 2. .125 = .5*.5*.5

Consider the random experiment of rolling a fair six-sided die once. What is the sample space of this random experiment?

1, 2, 3, 4, 5, 6

A fair coin is tossed two times. In a previous question, we determined that the outcomes in the sample space are equally likely to occur. Calculate the following probabilities (report all answers as a decimal to TWO decimal places). 1. P(both tosses result in heads landing face up) 2. P(one toss results in a head and the other toss results in a tails 3. P(both tosses result in tails landing face up)

1. .25 2. .50 3. .25

A family is planning on having three children. Is the family more likely to have all of one sex or two of one sex and one of the other? (We'll assume that it is equally likely that a child born is a girl or boy.) 1. How many outcomes are in the sample space? 2. Is the sex of newborn children independent? a) No. If the couple's first child is a boy (for example), the second child is more likely to be a girl. b) Yes since the sex of one child in the family does not depend on the sex of any child born to this couple previously. c) No. Genetics may play a role making it more likely that a child will be one particular sex than the other.

1. 8 - The possible outcomes are BBB, BBG, BGB, GBB, GGB, GBG, BGG, GGG 2. b) Yes since the sex of one child in the family does not depend on the sex of any child born to this couple previously.

An economist reported the following. State the type of inference (not an inference problem, estimation, or hypothesis test) for each statement. If the statement did not involve inference, choose "not an inference problem". 1. "Based on comparing a list of 40 common items between last month and this month, prices increased, on average." 2. "Based on a census of all adults between ages 18 and 65, unemployment was 3.8% this month." 3. "Based on a random sample of financial institutions, the average 30-year mortgage rate was 3.125% this month."

1. Hypothesis test - because two groups being compared based on sample of items 2. Not an inference problem - a census means that information is collected on all in the population. Therefore, no inference to the population needs to be made since we have information on the entire population already 3. Estimation - Data were collected from a sample to estimate the average 30-year mortgage rate for this month

For each scenario below, state if the given value is a statistic or parameter. 1. A fifth grade teacher wants to know what percent of his students read at least 30 minutes per night. He asks each student in his class if they read at least 30 minutes per night. 40% said they did. Is 40% a parameter or statistic? 2. An energy official wants to estimate the average oil output per well in the United States. From a random sample of 40 wells throughout the United States, the official obtains a mean of 10.1 barrels per day. Is 10.1 a parameter or statistic? 3. 40% of the 150 workers at a particular factory were paid less than $40,000 per year. You have the payroll data for all of the workers. Is 40% a parameter or statistic?

1. Parameter 2. Statistic 3. Parameter

For each scenario below, state if the given value is a statistic or parameter. 1. A researcher wants to estimate the average height of men aged 20 years or older. From a simple random sample of 40 women, the researcher obtains a mean height of 70.1 inches. Is 70.1 inches a parameter or statistic? 2. 85% of the 100 United States senators voted for a particular measure. Is 85% a parameter or statistic?

1. Statistic 2. Parameter

For each scenario below, state if the given value is a statistic or parameter. 1. It is recommended that children ages 2 to 18 consume fewer than 6 teaspoons of sugar (about 25 grams) each day. Suppose a nutritionist wanted to determine if the mean amount of sugar consumed by children aged 2 to 18 is indeed less than 6 teaspoons a day. From a random sample of 150 children aged 2 to 18, the nutritionist obtains a mean of 19 teaspoons of sugar consumed each day. Is 19 teaspoons per day a parameter or statistic? 2. 30% of a random sample of over 1000 dog owners poop scoop after their dog. Is 30% a parameter or statistic?

1. Statistic 2. Statistic

Three Probability Rules

1. The Multiplication Rule 2. The Addition Rule 3. The Complement Rule

You flip a coin two times. Let event A = both flips are heads Let event B = both flips are tails. 1. Are the two events disjoint? 2. Are the two events independent?

1. The two events are disjoint since the two flips cannot result in both two tails and two heads. 2. The two events are dependent since the probability of event B happening (both tails) would change if event A occurred.

Suppose one card is drawn from a deck of playing cards. Let event A = a club is drawn Let event B = an ace is drawn. 1. Are the two events disjoint? 2. Are the two events independent?

1. The two events are not disjoint since the card drawn can be both a club and an ace (the ace of clubs!) 2. The two events are independent since the probability of the card being an ace will be the same whether the card drawn is a club or any other suit

A family is planning on having three children. Is the family more likely to have all of one sex or two of one sex and one of the other? (We'll assume that it is equally likely that a child born is a girl or boy.) 1. True or False: The outcomes in the sample space are equally likely to occur. 2. What is the probability the family has 2 boys and 1 girl?

1. True - Because we're assuming it is equally likely to have a boy or girl on any one child, all 8 outcomes in the sample space would have the same probability of occurring: .125 2. .375 - From the probability distribution, each outcome in the sample space has a probability of .125. Since the outcomes in the sample space are disjoint, we can add the probabilities of these three outcomes: .125 + .125 + .125 = .375.

A family is planning on having three children. Is the family more likely to have all of one sex or two of one sex and one of the other? (We'll assume that it is equally likely that a child born is a girl or boy.) 1. Is the family more likely to have all of one sex or two of one sex and one of the other? a) both have the same chance of occurring b) two of one sex and one of the other (such as 2 girls and 1 boy) c) all of one sex (such as all girls) 2. What is the probability that at least one of the children will be a girl?

1. b) two of one sex and one of the other (such as 2 girls and 1 boy) P(2 girls) = P(2 boys) = 3/8, while P(all girls) = P(all boys) = 1/8 2. 1 - ((0.5)(0.5)(0.5) = 1 - .125 = .875

Suppose we flip a coin 10 times and obtain 8 heads. 1. What is a trial in this random experiment? 2. How many trials are there in this random experiment? 3. What is the outcome on each trial?

1. one flip of the coin 2. 10 3. the result of each flip of the coin (head or tails)

Sam surveys 150 adults who he believes are representative of all adults in a particular community. One hundred twenty say they are in favor of a certain issue on the ballot. Every adult in the sample has only two choices: favors the issue or does not favor the issue. Calculate the probability that a randomly selected person in this population will be in favor of this issue? Report your answer to ONE decimal place.

120 / 150 = .8

Consider the following process: A person flips a coin to determine if a balanced six sided die will be rolled. If the coin comes up heads the die will be rolled once and a number will be observed. If the coin comes up tails the die is not rolled and the experiment ends. How many possible outcomes are there for this random process?

7

Estimation Problem

Involves estimating the value of the parameter of interest based on the information we gather from the sample Ex. Our sample mean is the best guess as to the value of the population mean

Probability

Long-run proportion (or relative frequency) of times the outcome occurs when a random experiment is performed under identical conditions

Will scenarios that provide a parameter ever involve inference?

No - Since a parameter is value from a population, no inference is needed in a scenario that provides a parameter as we already know what the value in the population is. Inference is only performed when we don't know the population parameter.

A fifth grade teacher wants to know what percent of his students read at least 30 minutes per night. He asks each student in his class if they read at least 30 minutes per night. 40% said they did. Is this an inference problem?

No - The population is all students in this teacher's fifth-grade class. Since the teacher collected information on all in his population of interest, no inference is needed.

Two balanced six-sided dice are rolled at the same time (a red die and a green die). Let A = {doubles are rolled}. Calculate and report P(A). (Recall, A = doubles are rolled). Round your answer to FOUR decimal places.

Notice that 6 of the 36 outcomes result in doubles (1,1), (2,2), (3,3), (4,4), (5,5), (6,6). The equally likely formula can be used since each of the 36 outcomes in the sample space are equally likely to occur since both dice are "balanced". Therefore, P(A) = 6/36 = 1/6 = .1667

Trial

One repetition of a random experiment

An unbalanced coin is tossed two times. On this coin, the probability of getting a heads to land face up on any one toss is 0.4. There are four possible outcomes in the sample space (HH, HT, TH, TT), where HT (for example) means heads on first toss AND tails on second toss. Find the probability distribution (i.e. probability of each of these outcomes occurring). Report each to TWO decimal places. P(HH) P(TT) P(HT) P(TH)

P(HH) = .16 P(TT) = .36 P(HT) = .24 P(TH) = .24

Probability Formula:

P(outcome) = number of times the outcome occurs / total number of trials

Conditional Probabilities

Probabilities calculated based on whether or not a certain event already occurred

Hypothesis Test Problem

Questions of interest that make a claim about the population will involve doing a hypothesis test to test that claim. Often, the claim may be what a researcher wonders or believes to be true in the population. These claims will either: • mention or make reference to a particular value for the population parameter • question if there is a difference in the parameter being measured between two populations.

Outcome

Refers to the result of, or what happens on, a trial

A basketball player shoots two free throws in a row. Let A = she makes the first free throw. Let B = she misses the second free throw. Are the two events disjoint?

The two events are not disjoint since both of these events can happen in the same "experiment" (the two free throw attempts). That is, she can make the first attempt (event A) and miss the second attempt (event B).

Inference

To use data from a sample to make a generalization or conclusion about the population from which it came

An unbalanced coin is tossed two times. On this coin, the probability of getting a heads to land face up on any one toss is 0.4. There are four possible outcomes in the sample space (HH, HT, TH, TT), where HT (for example) means heads on first toss AND tails on second toss. True or False: The outcomes in the sample space are disjoint.

True - When tossing a coin twice, we cannot have both tosses be heads AND both tosses be tails

Disjoint

Two events are disjoint (or "mutually exclusive") if they both cannot occur in the same replication of a "situation"

The General Multiplication Rule:

When events are dependent, we can still multiply probabilities together. But, the probabilities multiplied together will be their conditioned probabilities.

A pharmaceutical company has developed a new drug they believe will help relieve symptoms associated with Crohn's Disease. A clinical trial involving 50 patients under the age of 30 with Crohn's Disease was conducted to compare the effectiveness of the new drug at relieving such symptoms compared to the standard drug. Is this scenario an example of inference or not?

Yes, It is an example of inference as the pharmaceutical company hopes to market the new drug to all patients with Crohn's Disease. Therefore, the patients in the clinical trial are a sample of all patients with Crohn's Disease.

A deck of playing cards has 52 cards, divided into four suits (diamonds, spades, clubs, and hearts). There are an equal number of cards in each suit in this deck. If three cards are drawn, what is the probability that the first two cards drawn are clubs and the last is red? There are three events in this problem: Let event A = first card drawn is a club Let event B = second card drawn is a club C = third card drawn is red. We want to find P(A and B and C) (i.e. the probability that the first card drawn is a club AND the second card drawn is a club AND the third card drawn is a red card). Suppose the first card drawn is a club and is not replaced into the deck before the second card is drawn. Also, suppose event A occurred (first card drawn is a club). What is the probability that the second card drawn is a club? a) 12/51 b) 12/52 c) 13/51 b) 13/52

a) 12/51

An unbalanced coin is tossed two times. On this coin, the probability of getting a heads to land face up on any one toss is 0.4. There are four possible outcomes in the sample space (HH, HT, TH, TT), where HT (for example) means heads on first toss AND tails on second toss. One method to find P(at least one head) is to apply the addition rule. How can the complement rule be used to find this probability? a) P(at least one head) = 1 - P(exactly one head) b) P(at least one head) = 1 - P(one head or less) c) P(at least one head) = 1 - P(no heads) d) P(at least one head) = 1 - P(both heads)

c) P(at least one head) = 1 - P(no heads, .36) = 0.64.

Consider the random experiment of selecting an envelope from a prize basket while blindfolded. The prize can either be monetary (like an envelope with a check) or a take-away prize (like an envelope with the title to a new car). Using the complement rule, what is the probability that at least one of the envelopes contains a take away prize? a) P(at least one take-away prize) = 1 - P(one take-away prize) b) P(one take-away prize) = P(at least one take-away prize) c) P(at least one take-away prize) = 1 - P(no take-away prize)

c) P(at least one take-away prize) = 1 - P(no take-away prize)

Which of the following most closely resembles the goal of simulating a random experiment? a) So that we can use technology that is complicated to control and implement b) So that we can repeat the experiment under different conditions over time c) So that the experiment can be easily replicated many times quickly

c) So that the experiment can be easily replicated many times quickly

An unbalanced coin is tossed two times. On this coin, the probability of getting a heads to land face up on any one toss is 0.4. There are four possible outcomes in the sample space (HH, HT, TH, TT), where HT (for example) means heads on first toss AND tails on second toss. What is the complement of "at least one head" in two tosses of the coin? a) exactly one head was tossed in the two tosses b) both tosses were heads c) neither toss was a head d) more than one of the above

c) neither toss was a head

Consider the random experiment of rolling one six-sided dice. Each side contains one number from 1 to 6 such that only one number appears on the dice. What is the trial? a) what lands face-up on one roll of the dice b) rolling the dice until a certain outcome appears c) one roll of the dice

c) one roll of the dice

Two events are ______ if they are the only two events in the sample space AND they cannot occur at the same time.

complements

Consider the random experiment of rolling one six-sided dice. Each side contains one number from 1 to 6 such that only one number appears on the dice. What are the possible outcomes on one trial? a) the number of times each value is rolled b) the number of times a certain value is rolled c) the number of times the die is rolled until a certain value (such as a 1) appears d) the value that is rolled (1, 2, 3, 4, 5, or 6) e) whether or not a certain value (such as a 1) is rolled

d) the value that is rolled (1, 2, 3, 4, 5, or 6)

Consider the random experiment of randomly selecting one number from 1 to 10. The event A = selecting a 5 and the event B = selecting a 2. Which of the following best describes the relationship between these two events, disjoint or complementary?

disjoint - since there is only one number being selected both a 5 and a 2 cannot be selected. Therefore, the events are disjoint. They are not complementary because although they cannot occur at the same time there are more than two possible events in the sample space

Once we have established that a scenario involves inference, the next step is to identify if the inference is an ______ problem or if it involves a ______ test.

estimation, hypothesis

A deck of playing cards has 52 cards, divided into four suits (diamonds, spades, clubs, and hearts). There are an equal number of cards in each suit in this deck. Two cards are drawn from the deck. Let event A = the first card drawn is a club. Let event B = the second card drawn is a club. If we sampled with replacement, are events A and B dependent or independent??

independent

Probability is the ______ proportion of times an outcome occurs

long-run

Anytime we have information on an entire population, ___ _____ is being made because we already know what is happening in the population

no inference

_____ and _____ both begin with the letter P, while _____ and _____ both begin with the letter S

parameter and population statistic and sample

A ______ is a number that describes some aspect of a population. A ______ is a number that is computed from data in a sample.

parameter, statistic

The ______ of all cases that have certain values of a variable is the same as the probability that a single case has a certain value

percent

Probabilities are ______, so we will write the percentage as a ______

proportions, proportion

Flipping a coin, rolling a die, drawing a card from a deck, and even taking a random sample, are examples of _____ _____ in the _____ sense

random experiments, probability

When randomly selecting people to be part of a sample, we're sampling ______ replacement

without - once a person is selected, they can't be selected again

A game show host selects participants from an audience of 60 people to participate in the game. For each round a different participant is randomly selected from the audience. Once an audience member from one row is selected no one from that row can be selected. Audience members are seated in rows of 10. What is the probability that an audience member in seat 25 is selected given an audience member in seat 23 was already selected.

zero

There are several key things to look for to tell if a scenario involves inference or not:

• Does the scenario mention a sample was taken? If so, there is a good chance the scenario involves making an inference. • Does the scenario mention a plan to estimate a parameter of interest? • Does the scenario make any sort of claim about a mean, median, or proportion? In such a situation, we may be wanting to make a decision about that claim in the population, which involves making an inference. • Does the scenario involve comparing two or more groups where the researchers want to make a comparison between these groups? If so, the scenario most likely involves making an inference to the populations.

The Monty Hall Problem

• Three closed doors were shown to the contestant. • The contestant knew that behind one door was a car, while behind the other two doors were goats. • The contestant would pick a door, hoping to win the car. • The host Monty Hall would reveal what was behind one of the two unpicked doors, which was always a goat. • Then he would ask the contestant whether they wanted to switch their choice to the remaining unopened door or to stick with the door he/she originally chose.

Probability Rule #2: The Addition Rule

If two events are disjoint (or "mutually exclusive), then the probability of one or the other occurring is the sum of their separate probabilities. • In other words, if event A and event B are disjoint, P(A or B) = P(A) + P(B), where "or" means either A occurs or B occurs.

Estimation

A best guess as to what the population problem is

Which form of technology would be best to use for simulating a random experiment?

A computer

Probability Distribution

A list (usually in table format) of all the possible outcomes and their probabilities Ex. outcome probability 2 heads .25 2 tails .25 1 head, 1 tail .50

Parameter

A number that describes some aspect of a population

Statistic

A number that is computed from data in a sample

Random Experiment

A process by which we observe something uncertain

Event

An outcome or list of several outcomes for which we want to find the probability of occurring

Probability Rule #1: The Multiplication Rule

If two events are independent, the probability of both occurring in the same "situation" can be found by multiplying the probabilities of each event together. • In other words, if event A and event B are independent, P(A and B) = P(A)*P(B), where "and" means both occur in the same "situation", and * means to multiply.

_____ letters near the ______ of the alphabet are generally used to notate an event of interest

Capital, beginning Ex. A = getting one head and one tail on two tosses of a coin

Suppose a market researcher wants to determine what proportion of teens have over 500 Facebook friends. The researcher cannot collect information on every teen, so he obtains a representative sample of all teens throughout the United States using random selection. From the data collected, he estimates the true proportion of teens in the United States who have over 500 Facebook friends. What is this an example of: Estimation Problem or Hypothesis Test Problem?

Estimation Problem

With the increasing popularity of online dating services the truthfulness of information in the personal profiles provided by users is a topic of interest. A study was performed to investigate misrepresentation of personal characteristics. In particular, researchers wondered what proportion of online daters believe they have misrepresented themselves in an online profile. They randomly sampled and contacted 1200 people with profiles in online dating services. Is this scenario an example of an estimation problem or hypothesis test problem?

Estimation Problem

A controversy exists over the distraction caused by digital billboards along highways. Researchers wondered if response time to road signs was greater when digital billboards were present compared to when they weren't present. In a study they conducted to answer their question of interest, 48 people made a 9 km/hr drive in a driving simulator. Drivers were instructed to change lanes according to roadside lane change signs. Some of the lane changes occurred near digital billboards and some did not. What was displayed on the digital billboard changed once during the time that the billboard was visible by the driver. Is this scenario an example of an estimation problem or hypothesis test problem?

Estimation Problem - The key is to identify that there are two groups (i.e. two populations) being compared: drivers who drive past digital billboards and drivers who do not drive past digital billboards

Independent Events

Events are independent if the occurrence of one event does not affect the occurrence of any of the other events • Another way to assess if two events are independent: if the probability of any event remains the same regardless of the occurrence of the other event, the two events are independent.

True or False? A simulated experiment should be done under different conditions than the original experiment to create randomness.

False

What is the sample space of a coin flip?

HH, TT, HT, TH

Researchers wondered if parents or their teen children use social media more, on average. A random sample of 1000 teens and their parents was taken. Each was asked how many times a day they check social media sites. Is this scenario an example of an estimation problem or hypothesis test problem?

Hypothesis Test

The Equally Likely Formula to calculate a probability:

If (and only if) the outcomes in the sample space are equally likely to occur, the following formula can be used to calculate the probability of an event of interest. Let A = an event of interest. P(A) = # of outcomes in the event of interest / number of outcomes in the sample space

Complement notation:

If A is an event, the complement of A is notated as A^c

Probability Rule #3: The Complement Rule

If two events are complements, the probability of the complement of an event = 1 - the probability of the event. • Notation: A^c = complement of A and means "NOT" A. In notation, P(A^c) = 1 - P(A)

Sample Space

The list of all possible outcomes in an experiment • The probability of, S, the sample space is 1; P(S) = 1

Law of Large Numbers

The long-run proportion of trials repeated under identical conditions will get closer to the true proportion as the number of trials increases

Sampling Without Replacement

The object selected is NOT replaced into the pool before the next object is selected, which means it cannot be selected again

Sampling with Replacement

The object selected is replaced into the "pool" before the next object is selected so that it's possible that object could be selected again

Equally Likely Outcomes

The outcomes in a sample space all have the same probability of occurring

Basic Probability Rules: The probability formula is a _____. The ______ of the fraction cannot be larger than the ______. Both the numerator and denominator are counts which can never be _____.

The probability formula is a fraction. The numerator of the fraction cannot be larger than the denominator. Both the numerator and denominator are counts which can never be negative.

Complement

The probability of an outcome not occurring • 1 − P(outcome)

A deck of playing cards has 52 cards, divided into four suits (diamonds, spades, clubs, and hearts). There are an equal number of cards in each suit in this deck. If three cards are drawn, what is the probability that the first two cards drawn are clubs and the last is red? There are three events in this problem: Let event A = first card drawn is a club Let event B = second card drawn is a club C = third card drawn is red. We want to find P(A and B and C) (i.e. the probability that the first card drawn is a club AND the second card drawn is a club AND the third card drawn is a red card). Suppose that the first card drawn is a club and is not replaced into the deck before the second card is drawn. Suppose also that the second card drawn is a club and is not replaced into the deck before the third card is drawn. What is the probability that the third card drawn is red? a) 26/50 b) 24/52 c) 24/50 d) 25/51 e) 1/2

a) 26/50 - The first two cards drawn are clubs, which are black cards. They are not replaced, so there are only 50 cards left in the deck. The deck started with 26 red cards and still has 26 red cards.

Allison wonders what the average starting salary for business majors in their first job out of college. She is obtains a random sample of recent business graduates from her school. This scenario is an example of which type of problem? a) Estimation Problem b) Hypothesis Test Problem

a) Estimation Problem

Consider the process of flipping a fair coin three times. What is the sample space of this random experiment? For the answer choices assume that H = heads and T = tails. a) HHH, HTH, THH, TTH, HHT, HTT, THT, TTT b) HHH, HTH, TTT, THT c) HHH, TTT, THH, TTH, HHT, HHH, TTT, THH

a) HHH, HTH, THH, TTH, HHT, HTT, THT, TTT • There are 2^3 = 8 possible outcomes since we have two outcomes on each flip and three flips. None of the outcomes should be repeated twice in our sample space.

Allison wonders what the average starting salary for business majors in their first job out of college. She is obtains a random sample of recent business graduates from her school. From the sample, Allison will estimate the average starting salary for all business majors in their first job out of college. Therefore, Allison is estimating a? a) Parameter b) Statistic

a) Parameter

A basketball player shoots two free throws in a row. Let A = she makes the first free throw. Let B = she misses the second free throw. What would be an argument that the two events are not independent? a) There may be a psychological effect - if she makes the first attempt, she may be more confident in her ability to make the second attempt. b) Both events can occur in one repetition of this "experiment". c) Exactly one of the events must occur in one repetition of this "experiment". d) The probability of making a free throw will remain the same regardless of whether or not she made the previous attempt.

a) There may be a psychological effect - if she makes the first attempt, she may be more confident in her ability to make the second attempt.

A political science professor wondered if there was a difference in the proportion favoring a recent U.S. Supreme Court Justice nominee between voters who identify themselves as Republican and voters who identify themselves as Democrats. She randomly sampled approximately 3000 Democrats and 3000 Republicans from voter registration cards for each party. Which is the following is correct? a) This is an inference problem that involves performing a hypothesis test. b) This is an inference problem that involves estimation only. c) This is not an inference problem.

a) This is an inference problem that involves performing a hypothesis test.

Match the scenario with the type (sampling with/without replacement) of sampling: a) Each card is put back into the deck before the next card is drawn b) Each card is not put back into the deck before the next card is drawn

a) sampling with replacement b) sampling without replacement

A deck of playing cards has 52 cards, divided into four suits (diamonds, spades, clubs, and hearts). There are an equal number of cards in each suit in this deck. If three cards are drawn, what is the probability that the first two cards drawn are clubs and the last is red? There are three events in this problem: Let event A = first card drawn is a club Let event B = second card drawn is a club C = third card drawn is red. We want to find P(A and B and C) (i.e. the probability that the first card drawn is a club AND the second card drawn is a club AND the third card drawn is a red card). Under what condition will these three events be independent of each other? a) Never as the probability that the third card drawn is a red will depend on the color of the first two cards drawn. b) As long as the cards are sampled with replacement, the three events will be independent. c) As long as the cards are sampled without replacement, the three events will be independent. d) Regardless of how the cards are sampled, the events will be independent because what is drawn on any one draw will not be affected by what was drawn on any previous draw.

b) As long as the cards are sampled with replacement, the three events will be independent.

Based on correctly simulating the Monty Hall problem, which strategy would you recommend to the contestant? a) stay b) switch c) it doesn't matter

b) switch - While it may not seem intuitive, a contestant will win the car 2/3 of the time they choose to switch but only 1/3 of the time they choose to stay. Another method of determining these probabilities will be shown later in this learning path.

One card is drawn from a deck of playing cards. Let A = {card drawn is a club}. What is the complement of event A a) A^c = the card drawn is the ace of diamonds b) A^c = the card drawn is red c) A^c = the card drawn is not a club d) A^c = the card drawn is a spade

c) A^c = the card drawn is not a club