Question 1

The number of goals scored at each game by a certain hockey team follows a Poisson distribution with a mean of 6 goals per game. Find the probability that the team will score more than three
Goals during a game.

Accepted Answer

The probability of scoring more than three goals in a game is found by subtracting the probability of scoring three or fewer goals from 1. The Poisson distribution formula for the probability of observing k events (goals in this case) is P(k; λ) = (e^(-λ) * λ^k) / k!, where λ is the average number of events (6 goals per game in this case), and k is the number of events of interest. To find the probability of scoring more than three goals, calculate the cumulative probability for 0, 1, 2, and 3 goals and subtract from 1. The correct answer is obtained by using the Poisson cumulative distribution function or calculating each probability individually and summing them up. The result is approximately 0.848796, indicating a high likelihood of scoring more than three goals in a game.

Question 2

Suppose the number of babies born each hour at a hospital follows a Poisson distribution with a mean of 7. Find the probability that exactly six babies will be born during a particular 1-hour
Period at this hospital.

Accepted Answer

The probability of exactly k events in a Poisson distribution is given by P(k; λ) = (e^(-λ) * λ^k) / k!, where λ is the mean number of events (7 in this case) and k is the number of events we're interested in (6 in this case). Plugging these values into the formula gives P(6; 7) = (e^(-7) * 7^6) / 6! ≈ 0.149003.

Question 3

Suppose the number of babies born each hour at a hospital follows a Poisson distribution with a mean of 3. Some people believe that the presence of a full moon increases the number of births that
Take place. Suppose during the presence of a full moon, the hospital experienced eight consecutive
Hours with more than four births each hour. Based on this fact, comment on the belief that the full
Moon increases the number of births.

Accepted Answer

The belief is supported because the probability of observing more than four births each hour for eight consecutive hours under the assumption of a Poisson distribution with a mean of 3 is extremely low (0.00000137). This suggests that such an event is highly unlikely to occur by chance, thus supporting the belief that the full moon could influence the number of births.

Question 4

The university police department must write, on average, five tickets per day to keep department revenues at budgeted levels. Suppose the number of tickets written per day follows a Poisson
Distribution with a mean of 8.9. Find the probability that exactly four tickets are written on a
Randomly selected day.

Accepted Answer

The answer of The university police department must write, on...

Question 5

The number of traffic accidents that occur on a particular stretch of road during a month
follows a Poisson distribution with a mean of 6.5. Find the probability that fewer than two
accidents will occur on this stretch of road during a month.

Accepted Answer

The answer of The number of traffic accidents that occur...

Question 6

According to a published study, 1 in every 6 men has been involved in a minor traffic accident. Suppose we have randomly and independently sampled twenty-five men and asked each whether
He has been involved in a minor traffic accident. How many of the 25 men do we expect to have
Never been involved in a minor traffic accident? Round to the nearest whole number.

Accepted Answer

Given that 1 in every 6 men has been involved in a minor traffic accident, it means 5 out of 6 men have never been involved in one. For 25 men, the expected number never involved in a minor traffic accident is (5/6) * 25 = 20.83, which rounds to 21.

Question 7

Suppose the number of babies born each hour at a hospital follows a Poisson distribution
with a mean of 4. Find the probability that exactly seven babies are born during a
randomly selected hour.

Accepted Answer

The answer of Suppose the number of babies born each...

Question 8

The number of road construction projects that take place at any one time in a certain city follows a Poisson distribution with a mean of 3. Find the probability that more than four road construction
Projects are currently taking place in the city.

Accepted Answer

The probability of more than four road construction projects occurring is found by subtracting the probability of four or fewer projects from 1. The Poisson distribution formula for the probability of exactly k events is (λ^k * e^(-λ))/k!, where λ is the mean number of events (3 in this case), e is the base of the natural logarithm, and k! is the factorial of k. To find the probability of more than four projects, calculate the cumulative probability for 0 to 4 projects and subtract from 1. The correct answer, after performing these calculations, is approximately 0.184737, indicating a relatively low probability of having more than four road construction projects simultaneously.

Question 9

It a recent study of college students indicated that 30% of all college students had at least one tattoo. A small private college decided to randomly and independently sample 15 of their students
And ask if they have a tattoo. Find the standard deviation for this binomial random variable.
Round to the nearest hundredth when necessary.

Accepted Answer

The standard deviation for a binomial random variable is given by the formula $\sqrt{n \cdot p \cdot (1-p)}$, where $n$ is the number of trials (in this case, 15 students), and $p$ is the probability of success (in this case, 0.30 for having at least one tattoo). Plugging in the values, we get $\sqrt{15 \cdot 0.30 \cdot (1-0.30)} = \sqrt{15 \cdot 0.30 \cdot 0.70} = \sqrt{3.15} \approx 1.77$.

Question 10

The university police department must write, on average, five tickets per day to keep department revenues at budgeted levels. Suppose the number of tickets written per day follows a Poisson
Distribution with a mean of 9.2. Find the probability that fewer than six tickets are written on a
Randomly selected day.

Accepted Answer

The probability of fewer than six tickets being written, when the number of tickets follows a Poisson distribution with a mean (λ) of 9.2, can be calculated using the Poisson probability formula. The correct answer involves calculating the sum of probabilities for 0, 1, 2, 3, 4, and 5 tickets, which results in approximately 0.104074.

Question 11

The number of traffic accidents that occur on a particular stretch of road during a month follows a Poisson distribution with a mean of 6.1. Find the probability that fewer than three accidents will
Occur next month on this stretch of road.

Accepted Answer

The probability of fewer than three accidents occurring next month on this stretch of road is given by the Poisson distribution with a mean of 6.1:P(X < 3) = P(X = 0) + P(X = 1) + P(X = 2)= e^(-6.1) * 6.1^0 / 0! + e^(-6.1) * 6.1^1 / 1! + e^(-6.1) * 6.1^2 / 2!= 0.057653

Question 12

A recent study suggested that 70% of all eligible voters will vote in the next presidential election. Suppose 20 eligible voters were randomly selected from the population of all eligible voters. How
Many of the sampled voters do we expect to vote in the next presidential election?

Accepted Answer

The expected number of voters who will vote is found by multiplying the total number of sampled voters (20) by the probability that any one of them will vote (70% or 0.7). So, 20 * 0.7 = 14.

Question 13

We believe that 81% of the population of all Business Statistics students consider statistics to be an exciting subject. Suppose we randomly and independently selected 39 students from the
Population. How many of the sampled students do we expect to consider statistics to be an
Exciting subject?

Accepted Answer

The answer of We believe that 81% of the population...

Question 14

The number of traffic accidents that occur on a particular stretch of road during a month follows a Poisson distribution with a mean of 7.4. Find the probability of observing exactly four accidents on
This stretch of road next month.

Accepted Answer

The probability of observing exactly $k$ accidents when the events follow a Poisson distribution with mean $\lambda$ is given by $P(X=k) = \frac{e^{-\lambda} \lambda^k}{k!}$. For $\lambda = 7.4$ and $k = 4$, the calculation is $P(X=4) = \frac{e^{-7.4} * 7.4^4}{4!}$, which results in approximately 0.076372.

Question 15

The number of goals scored at each game by a certain hockey team follows a Poisson distribution with a mean of 4 goals per game. Find the probability that the team scored exactly seven goals in
Each of two randomly selected games.

Accepted Answer

The probability of scoring exactly seven goals in a game, when the mean number of goals is 4, can be calculated using the Poisson distribution formula: P(X=k) = (λ^k * e^(-λ)) / k!, where λ is the mean number of goals (4 in this case), k is the number of goals for which we want to find the probability (7 in this case), and e is the base of the natural logarithm (approximately 2.71828). Plugging in the values, we get P(X=7) for one game. To find the probability of this happening in each of two randomly selected games, we square the probability for one game, since the games are independent.

Question 16

A small life insurance company has determined that on the average it receives 4 death
claims per day. Find the probability that the company receives at least seven death claims
on a randomly selected day.

Accepted Answer

The answer of A small life insurance company has determined...

Question 17

The number of traffic accidents that occur on a particular stretch of road during a month follows a Poisson distribution with a mean of 8.1. Find the probability that exactly three accidents will occur
On this stretch of road each of the next two months.

Accepted Answer

The probability of exactly three accidents occurring in a given month is calculated using the Poisson distribution formula: $P(X=k) = \frac{\lambda^k e^{-\lambda}}{k!}$, where $\lambda$ is the mean number of occurrences (8.1 in this case), $k$ is the number of occurrences for which the probability is being calculated (3 accidents), and $e$ is the base of the natural logarithm. Plugging in the values, we get the probability for one month. Since we are looking for the probability of this happening in each of the next two months, and assuming each month is independent, we square the probability for one month to get the probability for two months.

Question 18

The number of road construction projects that take place at any one time in a certain city follows a Poisson distribution with a mean of 4. Find the probability that exactly three road construction
Projects are currently taking place in this city.

Accepted Answer

The answer of The number of road construction projects that...

Question 19

The number of homeruns hit during a major league baseball game follows a Poisson distribution with a mean of 3.2. Find the probability that a randomly selected game would have exactly 5
Homeruns hit.

Accepted Answer

The probability of observing exactly $k$ events (in this case, 5 homeruns) in a Poisson distribution is given by the formula $P(X=k) = \frac{e^{-\lambda} \lambda^k}{k!}$, where $\lambda$ is the mean number of events (3.2 for this problem), and $k$ is the number of events of interest (5 homeruns). Plugging in the values, we get $P(X=5) = \frac{e^{-3.2} * 3.2^5}{5!}$, which calculates to approximately 0.114.

Question 20

An alarm company reports that the number of alarms sent to their monitoring center from customers owning their system follow a Poisson distribution with λ = 4.6 alarms per year. Find the
Probability that a randomly selected customer had more than 7 alarms reported.

Accepted Answer

The answer of An alarm company reports that the number...

Quiz 4: Discrete Random Variables

The number of goals scored at each game by a certain hockey team follows a Poisson distribution with a mean of 6 goals per game. Find the probability that the team will score more than three Goals during a game.

Suppose the number of babies born each hour at a hospital follows a Poisson distribution with a mean of 7. Find the probability that exactly six babies will be born during a particular 1-hour Period at this hospital.

The number of traffic accidents that occur on a particular stretch of road during a month follows a Poisson distribution with a mean of 6.5. Find the probability that fewer than two accidents will occur on this stretch of road during a month.

Suppose the number of babies born each hour at a hospital follows a Poisson distribution with a mean of 4. Find the probability that exactly seven babies are born during a randomly selected hour.

The number of road construction projects that take place at any one time in a certain city follows a Poisson distribution with a mean of 3. Find the probability that more than four road construction Projects are currently taking place in the city.

The number of traffic accidents that occur on a particular stretch of road during a month follows a Poisson distribution with a mean of 6.1. Find the probability that fewer than three accidents will Occur next month on this stretch of road.

A recent study suggested that 70% of all eligible voters will vote in the next presidential election. Suppose 20 eligible voters were randomly selected from the population of all eligible voters. How Many of the sampled voters do we expect to vote in the next presidential election?

We believe that 81% of the population of all Business Statistics students consider statistics to be an exciting subject. Suppose we randomly and independently selected 39 students from the Population. How many of the sampled students do we expect to consider statistics to be an Exciting subject?

The number of traffic accidents that occur on a particular stretch of road during a month follows a Poisson distribution with a mean of 7.4. Find the probability of observing exactly four accidents on This stretch of road next month.

The number of goals scored at each game by a certain hockey team follows a Poisson distribution with a mean of 4 goals per game. Find the probability that the team scored exactly seven goals in Each of two randomly selected games.

A small life insurance company has determined that on the average it receives 4 death claims per day. Find the probability that the company receives at least seven death claims on a randomly selected day.

The number of traffic accidents that occur on a particular stretch of road during a month follows a Poisson distribution with a mean of 8.1. Find the probability that exactly three accidents will occur On this stretch of road each of the next two months.

The number of road construction projects that take place at any one time in a certain city follows a Poisson distribution with a mean of 4. Find the probability that exactly three road construction Projects are currently taking place in this city.

The number of homeruns hit during a major league baseball game follows a Poisson distribution with a mean of 3.2. Find the probability that a randomly selected game would have exactly 5 Homeruns hit.

An alarm company reports that the number of alarms sent to their monitoring center from customers owning their system follow a Poisson distribution with λ = 4.6 alarms per year. Find the Probability that a randomly selected customer had more than 7 alarms reported.