Question 1

Which of the following statements is the most accurate about a Poisson random variable?

Accepted Answer

A Poisson random variable is used to count the number of events occurring within a specified time or space interval, such as the number of customers entering a store in an hour or the number of defective products in a batch. It is a discrete random variable, meaning it can only take on integer values.

Question 2

For a particular clothing store, a marketing firm finds that 16% of $10-off coupons delivered by mail are redeemed. Suppose six customers are randomly selected and are mailed $10-off coupons. What is the probability that three of the customers redeem the coupon?

Accepted Answer

This is a binomial probability problem where n = 6, p = 0.16, and we want the probability of k = 3 successes. The probability is calculated using the binomial formula: P(X = k) = C(n, k) * p^k * (1-p)^(n-k). Plugging in the values gives P(X = 3) = C(6, 3) * (0.16)^3 * (0.84)^3 = 0.0486.

Question 3

For a particular clothing store, a marketing firm finds that 16% of $10-off coupons delivered by mail are redeemed. Suppose six customers are randomly selected and are mailed $10-off coupons. What is the probability that no more than one of the customers redeems the coupon?

Accepted Answer

The answer of For a particular clothing store, a marketing...

Question 4

According to a study by the Centers for Disease Control and Prevention, about 33% of U.S. births are Caesarean deliveries. Suppose seven expectant mothers are randomly selected. The expected number of mothers who will not have a Caesarean delivery is ________.

Accepted Answer

The expected number of mothers who will not have a Caesarean delivery can be calculated by multiplying the probability of not having a Caesarean delivery (1 - 0.33 = 0.67) by the total number of mothers (7). So, 0.67 * 7 = 4.69.

Question 5

According to a study by the Centers for Disease Control and Prevention, about 33% of U.S. births are Caesarean deliveries. Suppose seven expectant mothers are randomly selected. What is the probability that at least one of the expectant mothers will have a Caesarean delivery?

Accepted Answer

The probability that at least one of the expectant mothers will have a Caesarean delivery can be found by subtracting the probability that none of them will have a Caesarean delivery from 1. The probability of not having a Caesarean delivery is 67% (100% - 33%). For seven mothers, this is (0.67)^7 ≈ 0.0606. Therefore, the probability of at least one Caesarean delivery is 1 - 0.0606 = 0.9394.

Question 6

For a particular clothing store, a marketing firm finds that 16% of $10-off coupons delivered by mail are redeemed. Suppose six customers are randomly selected and are mailed $10-off coupons. What is the expected number of coupons that will be redeemed?

Accepted Answer

The expected number of coupons redeemed can be found by multiplying the probability of a single coupon being redeemed (16% or 0.16) by the total number of coupons sent out (6). So, the expected number is 0.16 * 6 = 0.96.

Question 7

For a particular clothing store, a marketing firm finds that 16% of $10-off coupons delivered by mail are redeemed. Suppose six customers are randomly selected and are mailed $10-off coupons. What is the probability that at least two of the customers redeem the coupon?

Accepted Answer

The probability of at least two customers redeeming the coupon is 1 minus the probability of none or only one customer redeeming the coupon. The probability of none redeeming the coupon is (0.84)^6 = 0.2472. The probability of only one redeeming the coupon is 6 * (0.16) * (0.84)^5 = 0.3513. Therefore, the probability of at least two redeeming the coupon is 1 - 0.2472 - 0.3513 = 0.4015.

Question 8

Chauncey Billups, a current shooting guard for the Los Angeles Clippers, has a career free-throw percentage of 89.4%. Suppose he shoots six free throws in tonight's game. What is the probability that Billups makes five or more of his free throws?

Accepted Answer

The probability of making 5 or more free throws can be calculated using the binomial distribution. The probability of making exactly 5 free throws is 0.3632, and the probability of making exactly 6 free throws is 0.0511. Therefore, the probability of making 5 or more free throws is 0.3632 + 0.0511 = 0.4143.

Question 9

According to a Department of Labor report, the city of Detroit had a 20% unemployment rate in May. Eight working-age residents were chosen at random. What is the probability that at least two of the residents were unemployed?

Accepted Answer

The probability of at least two of the residents being unemployed is 1 - the probability of none of the residents being unemployed. The probability of none of the residents being unemployed is (1 - 0.2)^8 = 0.5033. Therefore, the probability of at least two of the residents being unemployed is 1 - 0.5033 = 0.4967.

Question 10

According to a study by the Centers for Disease Control and Prevention, about 33% of U.S. births are Caesarean deliveries. Suppose seven expectant mothers are randomly selected. What is the probability that at most two of the expectant mothers will have a Caesarean delivery?

Accepted Answer

The answer of According to a study by the Centers...

Question 11

According to a study by the Centers for Disease Control and Prevention, about 33% of U.S. births are Caesarean deliveries. Suppose seven expectant mothers are randomly selected. What is the standard deviation of the number of mothers who will have a Caesarean delivery?

Accepted Answer

The standard deviation for a binomial distribution is calculated using the formula √(n * p * (1-p)), where n is the number of trials (7 mothers), and p is the probability of success (0.33 for Caesarean deliveries). Plugging in the values, √(7 * 0.33 * 0.67) ≈ 1.24.

Question 12

Chauncey Billups, a current shooting guard for the Los Angeles Clippers, has a career free-throw percentage of 89.4%. Suppose he shoots six free throws in tonight's game. What is the expected number of free throws that Billups will make?

Accepted Answer

The expected number of free throws Billups will make is calculated by multiplying his free-throw percentage by the number of attempts. So, 0.894 (or 89.4%) times 6 equals approximately 5.364.

Question 13

On a particular production line, the likelihood that a light bulb is defective is 5%. Ten light bulbs are randomly selected. What are the mean and variance of the number of defective bulbs?

Accepted Answer

The answer of On a particular production line, the likelihood...

Question 14

Chauncey Billups, a current shooting guard for the Los Angeles Clippers, has a career free-throw percentage of 89.4%. Suppose he shoots six free throws in tonight's game. What is the standard deviation of the number of free throws that Billups will make?

Accepted Answer

The answer of Chauncey Billups, a current shooting guard for...

Question 15

The foreclosure crisis has been particularly devastating in housing markets in much of the south and west United States, but even when analysis is restricted to relatively strong housing markets the numbers are staggering. For example, in 2017 an average of three residential properties were auctioned off each weekday in the city of Boston, up from an average of one per week in 2011. What is the probability that exactly four foreclosure auctions occurred on a randomly selected weekday of 2017 in Boston?

Accepted Answer

This problem involves the Poisson distribution, which is used to model the number of rare events (in this case, foreclosure auctions) that occur in a fixed unit of time or space. The Poisson distribution has a single parameter, lambda, which represents the average number of events that occur in the unit of time or space.

In this problem, we are told that on average three residential properties were auctioned off each weekday in 2017 in Boston. Therefore, lambda = 3.

To find the probability that exactly four foreclosure auctions occurred on a randomly selected weekday of 2017 in Boston, we can use the Poisson probability mass function:

P(X = 4) = (e^-3)*(3^4)/4! = 0.1680

Therefore, the answer is choice A, 0.1680.

Question 16

According to a Department of Labor report, the city of Detroit had a 20% unemployment rate in May. Eight working-age residents were chosen at random. What is the probability that exactly one of the residents was unemployed?

Accepted Answer

The answer of According to a Department of Labor report,...

Question 17

According to a study by the Centers for Disease Control and Prevention, about 33% of U.S. births are Caesarean deliveries. Suppose seven expectant mothers are randomly selected. What is the probability that two of the expectant mothers will have a Caesarean delivery?

Accepted Answer

The answer of According to a study by the Centers...

Question 18

According to a Department of Labor report, the city of Detroit had a 20% unemployment rate in May. Eight working-age residents were chosen at random. What is the probability that exactly four residents were unemployed?

Accepted Answer

This is a binomial probability problem with n=8 trials, p=0.2 probability of success (i.e. being unemployed), and we want exactly 4 successes (i.e. 4 unemployed residents). We can use the binomial probability formula: P(X=k) = (n choose k) * p^k * (1-p)^(n-k), where (n choose k) = n!/((n-k)!k!). Plugging in the values, we get: P(X=4) = (8 choose 4) * 0.2^4 * 0.8^4 = 0.0459. Therefore, the probability that exactly four residents were unemployed is 0.0459, which corresponds to choice C.

Question 19

Chauncey Billups, a current shooting guard for the Los Angeles Clippers, has a career free-throw percentage of 89.4%. Suppose he shoots six free throws in tonight's game. What is the probability that Billups makes all six free throws?

Accepted Answer

The probability of Chauncey Billups making all six free throws is found by raising his free-throw percentage to the sixth power, since each shot is an independent event. Given his free-throw percentage is 89.4% or 0.894, the calculation is $0.894^6 \approx 0.5105$.

Question 20

According to a Department of Labor report, the city of Detroit had a 20% unemployment rate in May. Eight working-age residents were chosen at random. What was the expected number of unemployed residents, when eight working-age residents were randomly selected?

Accepted Answer

The expected number of unemployed residents can be calculated by multiplying the total number of residents chosen (8) by the probability of an individual being unemployed (20% or 0.2). Therefore, the expected number is 8 * 0.2 = 1.6.

Quiz 5: Discrete Probability Distributions

Which of the following statements is the most accurate about a Poisson random variable?

For a particular clothing store, a marketing firm finds that 16% of $10-off coupons delivered by mail are redeemed. Suppose six customers are randomly selected and are mailed $10-off coupons. What is the probability that three of the customers redeem the coupon?

For a particular clothing store, a marketing firm finds that 16% of $10-off coupons delivered by mail are redeemed. Suppose six customers are randomly selected and are mailed $10-off coupons. What is the probability that no more than one of the customers redeems the coupon?

According to a study by the Centers for Disease Control and Prevention, about 33% of U.S. births are Caesarean deliveries. Suppose seven expectant mothers are randomly selected. The expected number of mothers who will not have a Caesarean delivery is ________.

According to a study by the Centers for Disease Control and Prevention, about 33% of U.S. births are Caesarean deliveries. Suppose seven expectant mothers are randomly selected. What is the probability that at least one of the expectant mothers will have a Caesarean delivery?

For a particular clothing store, a marketing firm finds that 16% of $10-off coupons delivered by mail are redeemed. Suppose six customers are randomly selected and are mailed $10-off coupons. What is the expected number of coupons that will be redeemed?

For a particular clothing store, a marketing firm finds that 16% of $10-off coupons delivered by mail are redeemed. Suppose six customers are randomly selected and are mailed $10-off coupons. What is the probability that at least two of the customers redeem the coupon?

Chauncey Billups, a current shooting guard for the Los Angeles Clippers, has a career free-throw percentage of 89.4%. Suppose he shoots six free throws in tonight's game. What is the probability that Billups makes five or more of his free throws?

According to a Department of Labor report, the city of Detroit had a 20% unemployment rate in May. Eight working-age residents were chosen at random. What is the probability that at least two of the residents were unemployed?

According to a study by the Centers for Disease Control and Prevention, about 33% of U.S. births are Caesarean deliveries. Suppose seven expectant mothers are randomly selected. What is the probability that at most two of the expectant mothers will have a Caesarean delivery?

According to a study by the Centers for Disease Control and Prevention, about 33% of U.S. births are Caesarean deliveries. Suppose seven expectant mothers are randomly selected. What is the standard deviation of the number of mothers who will have a Caesarean delivery?

Chauncey Billups, a current shooting guard for the Los Angeles Clippers, has a career free-throw percentage of 89.4%. Suppose he shoots six free throws in tonight's game. What is the expected number of free throws that Billups will make?

On a particular production line, the likelihood that a light bulb is defective is 5%. Ten light bulbs are randomly selected. What are the mean and variance of the number of defective bulbs?

Chauncey Billups, a current shooting guard for the Los Angeles Clippers, has a career free-throw percentage of 89.4%. Suppose he shoots six free throws in tonight's game. What is the standard deviation of the number of free throws that Billups will make?

According to a Department of Labor report, the city of Detroit had a 20% unemployment rate in May. Eight working-age residents were chosen at random. What is the probability that exactly one of the residents was unemployed?

According to a study by the Centers for Disease Control and Prevention, about 33% of U.S. births are Caesarean deliveries. Suppose seven expectant mothers are randomly selected. What is the probability that two of the expectant mothers will have a Caesarean delivery?

According to a Department of Labor report, the city of Detroit had a 20% unemployment rate in May. Eight working-age residents were chosen at random. What is the probability that exactly four residents were unemployed?

Chauncey Billups, a current shooting guard for the Los Angeles Clippers, has a career free-throw percentage of 89.4%. Suppose he shoots six free throws in tonight's game. What is the probability that Billups makes all six free throws?

According to a Department of Labor report, the city of Detroit had a 20% unemployment rate in May. Eight working-age residents were chosen at random. What was the expected number of unemployed residents, when eight working-age residents were randomly selected?