Quant Methods Test 2
Which item matches the complement of A?
A^c
Recording whether an employee receives a raise .
Since an employee is unlikely to receive a raise right after he has just received a raise, the process is dependent.
Describe the outcomes of this random process as either independent or dependent. Briefly explain your choice. Recording the winning numbers in a specific lottery.
Since the winning numbers of one draw does not influence the next draw, the process is independent.
A market research assistant watches the next five customers as they leave a store. He records whether the customer is carrying a store bag that indicates the customer made a purchase. He writes down a yes or a no for each. Determine if the following statement is true or false. (A) The sample space S for this experiment has 10 elements
False, the sample space has 32 elements.
An administrator tracks absences among the staff working in an office. For each employee, define the events A={employee is absent} and S={employee is sick}. Mark the following statement True or False. If you believe that the statement is false, briefly say why you think it is false. The probability of an employee being absent is greater than the probability that the employee is absent given that the employee is sick.
False; it is possible that P(A)= P(A | S).
The probability of an employee being absent is greater than the probability that the employee is absent given that the employee is sick.
False; it is possible that P(A)= P(A | S).
Count the number of cars that pass by an intersection during consecutive five-minute periods on a highway leading into a city. Would these data allow us to use the law of large numbers eventually to learn P(more than 50 cars in five minutes)?
No, because the intensity of traffic would change over the time of day and the day of the week.
Which item matches independent events?
P(A and B) =P(A)P(B)
Which item matches the addition rule?
P(A or B)+P(A and B) = P(A)+P(B)
Consider the employees at a consulting firm and define the events E={the employee is an Engineer}, S={the employee is in a Secretarial position}, D={the employee has a college Degree}, and Ws={the employee has previous Work experience outside of this firm}. If P[S or D]=P[S]+P[D], what can we conclude about employees in secretarial positions and employees who have a college degree?
No secretaries have college degrees.
An insurer is studying the characteristics of those who buy its policies. It discovered that, among young drivers, 29% insure a foreign-made car. Among those who drive foreign-made cars, the insurer also discovered that 29% are young. Consider the events Y={randomly chosen driver is young} and F={randomly chosen driver insures foreign-made car}. Does the equality of these percentages mean that Y and F are independent events?
No. The statement of the question first gives P(F|Y) is 0.29 and then P(Y|F) is 0.29. In order to be independent P(F|Y)=P(F), but P(F) is not given.
An insurer is studying the characteristics of those who buy its policies. It discovered that, among young drivers, 44% insure a foreign-made car. Among those who drive foreign-made cars, the insurer also discovered that 44% are young. Consider the events Y={randomly chosen driver is young} and F={randomly chosen driver insures foreign-made car}. Does the equality of these percentages mean that Y and F are independent events?
No. The statement of the question first gives P(F|Y) is 0.44 and then P(Y|F) is 0.44. In order to be independent P(F|Y)=P(F), but P(F) is not given.
Probability of B given A
P(A and B) P(A)
Which item matches Boole's inequality?
P(A or B) ≤ P(A)+P(B)
Which of the following implications comes from the law of large numbers?
Proportions get close to probabilities in the long run.
A brand of men's pants offered for sale at a clothing store comes in various sizes. The possible waist and inseam (length of the pant leg) sizes are given below. Waist: {24 inches, 26 inches, ..., 46 inches} Inseam: {28 inches, 30 inches, ..., 40 inches} Define the event A={waist 40 inches or larger40 inches or larger} and B={inseam 36 inches or larger36 inches or larger}. Complete parts (a) through (c) below. (A) Describe a customer whose choice is in the event (A and B). (B) What would it mean if P(A and B)equals=P(A)times×P(B)? (C) Does the choice of a talltall customer with a thinthin waist in the event (A and B), the event (A or B), both, or neither?
(A) A customer shopping for a tall man with a large waist (B) The choice of waist size is independent of the length of the pant leg. (C) Neither
A shopper in a convenience store can make a food selection from fresh foods, refrigerated packages, deli items, or frozen items. Let the event A equals={fresh, refrigerated, deli}, and B equals={frozen, refrigerated}. (a) Find the intersection of A and B. (b) Find the union of A and B. (c) Find the event A
(A) Refrigerated (B) Fresh, Refrigerated, Deli, Frozen (C) Forzen
A recent study looked into the amount of debt accumulated by recent college graduates. The study found that, among those with student loans, 36% said working during college affected their grades. (a) Convert this statement into a conditional probability, including a short description of the associated sample space. (b) What percentage of students working during college have student loans? Can you tell? (a) Let R={recent college graduate}, H={has student loans}, and W={working affected grades}.
(A) The conditional probability is P(W|H) = 36%. The sample space is R . (B) There is not enough information to know the percentage of students working during college that have student loans
A recent study looked into the amount of debt accumulated by recent college graduates. The study found that, among those with student loans, 33% said working during college affected their grades. (a) Convert this statement into a conditional probability, including a short description of the associated sample space. (b) What percentage of students working during college have student loans? Can you tell? (a) Let R={recent college graduate}, H={has student loans}, and W={working affected grades}. (B) Select the correct choice and, if necessary, fill in the answer box to complete your choice.
(A) The conditional probability is P(W|H)=33%. The sample space is R (B) There is not enough information to know the percentage of students working during college that have student loans .
A company claims that 30% of its candies are blue, 20% are orange, 16% are green, 14% are yellow, and 10% each red and brown. Complete parts a and b below. (A) Which statement below describes the sample space? (B) The probability that the candy is blue or red is (C) The probability that the candy is not green is (D) Pick three candies in a row randomly from three separate packages. (E) Which statement below describes the sample space? (F) The probability that all three candies picked are blue is (G) The probability that the third candy picked is red is (H) The probability that at least one of the three candies is blue is
(A) The six possible colors of the candy picked (B) .40 (C) .84 (E) The 216 possible color combinations of the candies picked. (F) .027 (G) .1 (H) .657
In the weeks following a crash, airlines often report a drop in the number of passengers, probably because people are wary of flying because they just learned of an accident. (a) A travel agent suggests that, since the law of large numbers makes it highly unlikely to have two plane crashes within a few weeks of each other, flying soon after a crash is the safest time. What do you think? (b) If the airline industry proudly announces that it has set a new record for the longest period of safe flights, would you be reluctant to fly? Are the airlines due to have a crash? (a) If there was a crash recently, is another crash unlikely? (b) If there has not been a crash for a long time, is a crash more likely?
(A) No, the law of large numbers only applies to the long run proportion. (B) No, a random process does not compensate for what has happened in the past.
A Web site recorded whether visitors click on a shown ad. The following plot shows the outcomes for a sequence of 100 visitors, with a 1 shown if the visitor clicked on the ad and a 0 otherwise. Complete parts (a) and (b) below. (a) Does it appear that the Law of Large Numbers is applicable if this sequence continues indefinitely? (b) Can we find the probability of clicking on the shown ad from looking at these 100 observations?
(A) Yes, because the data do not appear to have a pattern. (B) No, because the number of observed outcomes is not very large.
Consider the employees at a consulting firm and define the events E={the employee is an Engineer}, S={the employee is in a Secretarial position}, D={the employee has a college Degree}, and W={the employee has previous Work experience outside of this firm}. Which of the following represents an employee that is an engineer without previous work experience outside of this firm?
(E and W ^c)
Which item matches the union?
A or B
It's time for an advertising firm to renew its contract with a client. The advertising firm wants the client to increase the amount it spends to advertise. If the firm proposes to continue the current contract, there's a 30% chance that the client will accept the proposal. To increase the business will require a second proposal that has a 60% chance of approval. Alternatively, the advertising firm can begin the negotiations by proposing an elaborate advertising campaign. If it takes this approach, there's a 20% chance that the client will approve the expanded proposal without needing a second proposal. Which approach should the advertising firm take if it wants to grow the business? Identify any assumptions you've made.
Assuming that the events of the client accepting the proposal to renew the current contract and the client accepting the second proposal are independent, the advertising firm should adopt the approach involving the single proposal the single proposal because the chance of growing the business with the two-proposal approach is 18%, which is less than the chance of growing the business with the single-proposal approach.
A basketball team is down by 2 points with only a few seconds remaining in the game. There is a 70% chance that the team will be able to make a 2-point shot and tie the game, compared to a 30% chance that it will make a 3-point shot and win. In overtime, the team has a 30% chance of winning. What should the coach do, go for the 2-point shot or the 3-point shot? Be sure to identify any assumptions.
Assuming that the outcomes are independent , the coach should have the team go for the 3-point shot. The chance of making the 2-point shot and winning in overtime is 21%, which is less than the chance of making the 3-point shot.
If two events A and B are independent, which of the following is not a true statement about A and B?
Events A and B cannot occur at the same time.
The Human Resources (HR) group at a large accounting firm interviews prospective candidates for new hires. After each interview, the firm rates the candidate on a 10-point scale, with the rating 10 denoting exceptionally good candidates and 1 denoting those that the firm rates poor. The HR group rated 6 candidates on Monday and 6 candidates on Tuesday. The outcomes of these 12 ratings form the sample space. Determine if the following statement is true or false. The events A={3 candidates on Monday rate above 7} and B={two candidates on Tuesday rate above 7} are disjoint events.
False, both events could happen.
A company seeks to hire engineering graduates who also speak a foreign language. Should you describe the combination of talents as an intersection or a union?
The combination of talents should be described as intersection.
A market research assistant watches the next five customers as they leave the store. He records whether the customer is carrying a store bag that indicates the customer made a purchase. He writes down a yes or a no for each. Define the events, A, B, and C, as shown below. A={first two shoppers have a bag} B={last two shoppers have a bag} C={last three shoppers have a bag} Mark the statement given below true or false and explain your reasoning. P(A)+P(B)=P(A or B)
The given statement is false because A and B are not necessarily disjoint events.
Multiplication Rule
The joint probability of two events A and B is the product of the marginal probability of one times the conditional probability of the other. This is known as the multiplication rule. P(A and B) = P(A) x P(B/A)
The Law of Large Numbers
The law of large numbers states that the relative frequency of an outcome converges to a number, the probability of the outcome, as the number of observed outcomes increases. The law of large numbers applies to data that do not have a pattern. Identify any scatterplots that do not have a strong pattern.
Eighty percent of customers at the snack counter of a movie theater buy drinks. Among those who buy drinks, 15% also purchase popcorn. What's the probability that a customer at the counter buys a drink and popcorn? Theaters use this type of calculation to decide which products should be bundled to appeal to customers.
The probability is .12. (Type an integer or a decimal.)
What probability rule identifies dependent events?
The probability rule that identifies dependent events is P(A)≠P(A|B).
What probability rule identifies independent events?
The probability rule that identifies independent events is P(A)=P(A|B).
An administrator tracks absences among the staff working in an office. For each employee, A={employee is absent}, and S={employee is sick}. If Upper A1 is independent of Upper A2, then finding out that Employee 1 is absent increases the chance that Employee 2 is absent.
This statement is false. Independence implies that one event does not influence the probability of the other.
If A 1is independent of A2, P(A 1,A2)= P(A2)
The statement is false because the expression should be P(A1 /A2) = P( A1).
The Human Resources (HR) group at a large accounting firm interviews prospective candidates for new hires. After each interview, the firm rates the candidate on a 10-point scale, with the rating 10 denoting exceptionally good candidates and 1 denoting those that the firm rates poor. The HR group rated 6 candidates on Monday and 6 candidates on Tuesday. The outcomes of these 12 ratings form the sample space. Mark the statement given below true or false and explain your reasoning. The HR group has monitored the outcome of these interviews for several years. The Law of Large Numbers assures us that HR personnel can use these data to learn the probability of a candidate scoring above 8 during an interview.
The statement is false because the relative frequency tends to the probability in the long run only if the data lack patterns.
An administrator tracks absences among the staff working in an office. For each employee, A ={employee is absent}, and S={employee is sick}. If the chance for an employee to be absent is greater than the chance for an employee to be sick, then A and S are dependent events.
The statement is false. The fact that Upper P ( A) right is greater than P(S) does not imply dependence.
If an employee is picked at random from the clerical employees, then P(S|C) is the probability that this employee makes more than $120,000 annually.
The statement is true. The expression P(S|C) is the probability that the person makes $120,000 annually, given that they are from the clerical employees.
Independence of S with each of A, C, and M implies that an equal proportion of employees within these categories makes above $120,000 annually.
The statement is true. Two events are independent if the outcome of one of those events does not influence the outcome of the other. Therefore, if any of A, C, or M are true, it should not effect whether S is true. This means that there needs to be an equal probability of a randomly chosen employee from any category making more than $120,000 annually.
Intersection or Union
The union of two events A and B is the collection of outcomes in A, in B, or in both. The intersection of two events A and B is the event consisting of the outcomes in both A and B. Since the company is seeking people who have an engineering degree and speak a foreign language, the combination should be described as an intersection.
A market research assistant watches the next five customers as they leave a store. He records whether the customer is carrying a store bag that indicates the customer made a purchase. He writes down a yes or a no for each. Determine if the following statement is true or false. The probability that a randomly chosen customer purchases with a credit card or spends more than $50 is the same as or larger than the probability that the customer purchases with a credit card and spends more than $50.
True
The human resources division classifies employees of a firm into one of three categories: administrative, clerical, or management. Suppose we choose an employee at random. Define the events as Aequals={administrative}, Cequals={clerical}, and Mequals={management}. Event A occurs, for example, if the randomly chosen employee is an administrator. Event S occurs if the randomly chosen employee (from any category) makes more than $120,000 annually. Mark the statement below as true or false. If you believe that a statement is false, briefly say why you think it is false. Independence of S with each of A, C, and M implies that an equal proportion of employees within these categories makes above $120,000 annually.
True
