The sample mean = 2.50 and the sample standard deviation = 0.8302. The distribut
ID: 2948981 • Letter: T
Question
The sample mean = 2.50 and the sample standard deviation = 0.8302. The distribution can be written as X ~ U(1.5, 4.5).
a. What type of distribution is this?
b. In this distribution, outcomes are equally likely. What does this mean?
c. What is the height of f(x) for the continuous probability distribution?
d. What are the constraints for the values of x?
e. Graph P(2 < x < 3).
f. What is P(2 < x < 3)?
g. What is P(x < 3.5| x < 4)?
h. What is P(x = 1.5)? 24. What is the 90th percentile of square footage for homes?
i. Find the probability that a randomly selected home has more than 3,000 square feet given that you already know the house has more than 2,000 square feet.
18. A customer service representative must spend different amounts of time with each customer to resolve various concerns. The amount of time spent with each customer can be modeled by the following distribution: X ~ Exp(0.2)
a. What type of distribution is this?
b. Are outcomes equally likely in this distribution? Why or why not?
c. What is m? What does it represent?
d. What is the mean?
e. What is the standard deviation?
f. State the probability density function.
g. Graph the distribution.
h. Find P(2 < x < 10).
i. Find P(x > 6).
j. Find the 70th percentile.
Chapter 4 & 5 – Computation Practice
1. A baker is deciding how many batches of muffins to make to sell in his bakery. He wants to make enough to sell every one and no fewer. Through observation, the baker has established a probability distribution.
a. Define the random variable X.
b. What is the probability the baker will sell more than one batch? P(x > 1) = _______
c. What is the probability the baker will sell exactly one batch? P(x = 1) = _______
d. On average, how many batches should the baker make?
2. Javier volunteers in community events each month. He does not do more than five events in a month. He attends exactly five events 35% of the time, four events 25% of the time, three events 20% of the time, two events 10% of the time, one event 5% of the time, and no events 5% of the time.
a. Define the random variable X.
b. What values does x take on?
c. Construct a PDF table.
d. Find the probability that Javier volunteers for less than three events each month. P(x < 3) = _______
e. Find the probability that Javier volunteers for at least one event each month. P(x > 0) = _______
3. You are playing a game by drawing a card from a standard deck and replacing it. If the card is a face card, you win $30. If it is not a face card, you pay $2. There are 12 face cards in a deck of 52 cards. What is the expected value of playing the game?
4. You are playing a game by drawing a card from a standard deck and replacing it. If the card is a face card, you win $30. If it is not a face card, you pay $2. There are 12 face cards in a deck of 52 cards. Should you play the game?
5. A theater group holds a fund-raiser. It sells 100 raffle tickets for $5 apiece. Suppose you purchase four tickets. The prize is two passes to a Broadway show, worth a total of $150. a. What are you interested in here? b. In words, define the random variable X. c. List the values that X may take on. d. Construct a PDF. e. If this fund-raiser is repeated often and you always purchase four tickets, what would be your expected average winnings per raffle?
6. You buy a lottery ticket to a lottery that costs $10 per ticket. There are only 100 tickets available to be sold in this lottery. In this lottery there are one $500 prize, two $100 prizes, and four $25 prizes. Find your expected gain or loss.
7. People visiting video rental stores often rent more than one DVD at a time. The probability distribution for DVD rentals per customer at Video To Go is given in the following table. There is a five-video limit per customer at this store, so nobody ever rents more than five DVDs.
a. Describe the random variable X in words.
b. Find the probability that a customer rents three DVDs.
c. Find the probability that a customer rents at least four DVDs.
d. Find the probability that a customer rents at most two DVDs. Another shop, Entertainment Headquarters, rents DVDs and video games. The probability distribution for DVD rentals per customer at this shop is given as follows. They also have a five-DVD limit per customer.
e. At which store is the expected number of DVDs rented per customer higher?
f. If Video to Go estimates that they will have 300 customers next week, how many DVDs do they expect to rent next week? Answer in sentence form.
g. If Video to Go expects 300 customers next week, and Entertainment HQ projects that they will have 420 customers, for which store is the expected number of DVD rentals for next week higher? Explain.
h. Which of the two video stores experiences more variation in the number of DVD rentals per customer? How do you know that?
8. The Higher Education Research Institute at UCLA collected data from 203,967 incoming first-time, full-time freshmen from 270 four-year colleges and universities in the 71.3% of those students replied that, yes, they believe that same-sex couples should have the right to legal marital status. Suppose that you randomly pick eight first-time, full-time freshmen from the survey. You are interested in the number that believes that same sex-couples should have the right to legal marital status.
a. In words, define the random variable X.
b. X ~ _____(_____,_____)
c. What values does the random variable X take on?
d. Construct the probability distribution function (PDF).
e. On average (?), how many would you expect to answer yes?
f. What is the standard deviation (?)?
g. What is the probability that at most five of the freshmen reply “yes”?
h. What is the probability that at least two of the freshmen reply “yes”?
9. More than 96 percent of the very largest colleges and universities (more than 15,000 total enrollments) have some online offerings. Suppose you randomly pick 13 such institutions. We are interested in the number that offer distance learning courses.
a. In words, define the random variable X.
b. List the values that X may take on.
c. Give the distribution of X. X ~ _____(_____,_____)
d. On average, how many schools would you expect to offer such courses?
e. Find the probability that at most ten offer such courses.
f. Is it more likely that 12 or that 13 will offer such courses? Use numbers to justify your answer numerically and answer in a complete sentence.
10. At The Fencing Center, 60% of the fencers use the foil as their main weapon. We randomly survey 25 fencers at The Fencing Center. We are interested in the number of fencers who do not use the foil as their main weapon.
a. In words, define the random variable X.
b. List the values that X may take on.
c. Give the distribution of X. X ~ _____(_____,_____)
d. How many are expected to not to use the foil as their main weapon?
e. Find the probability that six do not use the foil as their main weapon.
f. Based on numerical values, would you be surprised if all 25 did not use foil as their main weapon? Justify your answer numerically
11. The chance of an IRS audit for a tax return with over $25,000 in income is about 2% per year. We are interested in the expected number of audits a person with that income has in a 20-year period. Assume each year is independent.
a. In words, define the random variable X.
b. List the values that X may take on.
c. Give the distribution of X. X ~ _____(_____,_____)
d. How many audits are expected in a 20-year period?
e. Find the probability that a person is not audited at all.
f. Find the probability that a person is audited more than twice
12. The literacy rate for a nation measures the proportion of people age 15 and over that can read and write. The literacy rate in Afghanistan is 28.1%. Suppose you choose 15 people in Afghanistan at random. Let X = the number of people who are literate.
a. Sketch a graph of the probability distribution of X.
b. Using the formulas, calculate the (i) mean and (ii) standard deviation of X.
c. Find the probability that more than five people in the sample are literate. Is it is more likely that three people or four people are literate.
13. Births are approximately uniformly distributed between the 52 weeks of the year. They can be said to follow a uniform distribution from one to 53 (spread of 52 weeks).
a. Graph the probability distribution.
b. ? = _________
c. ? = _________
d. Find the probability that a person is born at the exact moment week 19 starts. That is, find P(x = 19)
e. P(2 < x < 31) = _________
f. Find the probability that a person is born after week 40.
g. P(12 < x|x < 28) = _________
h. Find the 70th percentile.
i. Find the minimum for the upper quarter.
14. A random number generator picks a number from one to nine in a uniform manner. a. Graph the probability distribution.
b. ? = _________
c. ? = _________
d. P(3.5 < x < 7.25) = _________
e. P(x > 5.67)
f. Find the 90th percentile.
15. A subway train on the Red Line arrives every eight minutes during rush hour. We are interested in the length of time a commuter must wait for a train to arrive. The time follows a uniform distribution.
a. Define the random variable. X = _______
b. X ~ _______
c. Graph the probability distribution.
d. f(x) = _______
e. ? = _______
f. ? = _______
g. Find the probability that the commuter waits less than one minute.
h. Find the probability that the commuter waits between three and four minutes. i. Sixty percent of commuters wait more than how long for the train? State this in a probability question, similarly to parts g and h, draw the picture, and find the probability.
16. Suppose that the longevity of a light bulb is exponential with a mean lifetime of eight years.
a. Find the probability that a light bulb lasts less than one year.
b. Find the probability that a light bulb lasts between six and ten years.
c. Seventy percent of all light bulbs last at least how long?
17. The cost of all maintenance for a car during its first year is approximately exponentially distributed with a mean of $150.
a. Define the random variable. X = _________________________________.
b. X ~ = ________
c. ? = ________
d. ? = ________
e. Draw a graph of the probability distribution. Label the axes.
f. Find the probability that a car required over $300 for maintenance during its first year.
18. Suppose that the useful life of a particular car battery, measured in months, decays with parameter 0.025. We are interested in the life of the battery.
a. Define the random variable. X = _________________________________.
b. Is X continuous or discrete?
c. X ~ ________
d. On average, how long would you expect one car battery to last?
e. On average, how long would you expect nine car batteries to last, if they are used one after another?
f. Find the probability that a car battery lasts more than 36 months.
g. Seventy percent of the batteries last at least how long?
19. A web site experiences traffic during normal working hours at a rate of 12 visits per hour. Assume that the duration between visits has the exponential distribution.
a. Find the probability that the duration between two successive visits to the web site is more than ten minutes.
b. The top 25% of durations between visits are at least how long?
c. Suppose that 20 minutes have passed since the last visit to the web site. What is the probability that the next visit will occur within the next 5 minutes?
d. Find the probability that less than 7 visits occur within a one-hour period.
The sample mean = 2.50 and the sample standard deviation = 0.8302. The distribution can be written as X ~ U(1.5, 4.5).
a. What type of distribution is this?
b. In this distribution, outcomes are equally likely. What does this mean?
c. What is the height of f(x) for the continuous probability distribution?
d. What are the constraints for the values of x?
e. Graph P(2 < x < 3).
f. What is P(2 < x < 3)?
g. What is P(x < 3.5| x < 4)?
h. What is P(x = 1.5)? 24. What is the 90th percentile of square footage for homes?
i. Find the probability that a randomly selected home has more than 3,000 square feet given that you already know the house has more than 2,000 square feet.
18. A customer service representative must spend different amounts of time with each customer to resolve various concerns. The amount of time spent with each customer can be modeled by the following distribution: X ~ Exp(0.2)
a. What type of distribution is this?
b. Are outcomes equally likely in this distribution? Why or why not?
c. What is m? What does it represent?
d. What is the mean?
e. What is the standard deviation?
f. State the probability density function.
g. Graph the distribution.
h. Find P(2 < x < 10).
i. Find P(x > 6).
j. Find the 70th percentile.
Chapter 4 & 5 – Computation Practice
1. A baker is deciding how many batches of muffins to make to sell in his bakery. He wants to make enough to sell every one and no fewer. Through observation, the baker has established a probability distribution.
a. Define the random variable X.
b. What is the probability the baker will sell more than one batch? P(x > 1) = _______
c. What is the probability the baker will sell exactly one batch? P(x = 1) = _______
d. On average, how many batches should the baker make?
2. Javier volunteers in community events each month. He does not do more than five events in a month. He attends exactly five events 35% of the time, four events 25% of the time, three events 20% of the time, two events 10% of the time, one event 5% of the time, and no events 5% of the time.
a. Define the random variable X.
b. What values does x take on?
c. Construct a PDF table.
d. Find the probability that Javier volunteers for less than three events each month. P(x < 3) = _______
e. Find the probability that Javier volunteers for at least one event each month. P(x > 0) = _______
3. You are playing a game by drawing a card from a standard deck and replacing it. If the card is a face card, you win $30. If it is not a face card, you pay $2. There are 12 face cards in a deck of 52 cards. What is the expected value of playing the game?
4. You are playing a game by drawing a card from a standard deck and replacing it. If the card is a face card, you win $30. If it is not a face card, you pay $2. There are 12 face cards in a deck of 52 cards. Should you play the game?
5. A theater group holds a fund-raiser. It sells 100 raffle tickets for $5 apiece. Suppose you purchase four tickets. The prize is two passes to a Broadway show, worth a total of $150. a. What are you interested in here? b. In words, define the random variable X. c. List the values that X may take on. d. Construct a PDF. e. If this fund-raiser is repeated often and you always purchase four tickets, what would be your expected average winnings per raffle?
6. You buy a lottery ticket to a lottery that costs $10 per ticket. There are only 100 tickets available to be sold in this lottery. In this lottery there are one $500 prize, two $100 prizes, and four $25 prizes. Find your expected gain or loss.
7. People visiting video rental stores often rent more than one DVD at a time. The probability distribution for DVD rentals per customer at Video To Go is given in the following table. There is a five-video limit per customer at this store, so nobody ever rents more than five DVDs.
a. Describe the random variable X in words.
b. Find the probability that a customer rents three DVDs.
c. Find the probability that a customer rents at least four DVDs.
d. Find the probability that a customer rents at most two DVDs. Another shop, Entertainment Headquarters, rents DVDs and video games. The probability distribution for DVD rentals per customer at this shop is given as follows. They also have a five-DVD limit per customer.
e. At which store is the expected number of DVDs rented per customer higher?
f. If Video to Go estimates that they will have 300 customers next week, how many DVDs do they expect to rent next week? Answer in sentence form.
g. If Video to Go expects 300 customers next week, and Entertainment HQ projects that they will have 420 customers, for which store is the expected number of DVD rentals for next week higher? Explain.
h. Which of the two video stores experiences more variation in the number of DVD rentals per customer? How do you know that?
8. The Higher Education Research Institute at UCLA collected data from 203,967 incoming first-time, full-time freshmen from 270 four-year colleges and universities in the 71.3% of those students replied that, yes, they believe that same-sex couples should have the right to legal marital status. Suppose that you randomly pick eight first-time, full-time freshmen from the survey. You are interested in the number that believes that same sex-couples should have the right to legal marital status.
a. In words, define the random variable X.
b. X ~ _____(_____,_____)
c. What values does the random variable X take on?
d. Construct the probability distribution function (PDF).
e. On average (?), how many would you expect to answer yes?
f. What is the standard deviation (?)?
g. What is the probability that at most five of the freshmen reply “yes”?
h. What is the probability that at least two of the freshmen reply “yes”?
9. More than 96 percent of the very largest colleges and universities (more than 15,000 total enrollments) have some online offerings. Suppose you randomly pick 13 such institutions. We are interested in the number that offer distance learning courses.
a. In words, define the random variable X.
b. List the values that X may take on.
c. Give the distribution of X. X ~ _____(_____,_____)
d. On average, how many schools would you expect to offer such courses?
e. Find the probability that at most ten offer such courses.
f. Is it more likely that 12 or that 13 will offer such courses? Use numbers to justify your answer numerically and answer in a complete sentence.
10. At The Fencing Center, 60% of the fencers use the foil as their main weapon. We randomly survey 25 fencers at The Fencing Center. We are interested in the number of fencers who do not use the foil as their main weapon.
a. In words, define the random variable X.
b. List the values that X may take on.
c. Give the distribution of X. X ~ _____(_____,_____)
d. How many are expected to not to use the foil as their main weapon?
e. Find the probability that six do not use the foil as their main weapon.
f. Based on numerical values, would you be surprised if all 25 did not use foil as their main weapon? Justify your answer numerically
11. The chance of an IRS audit for a tax return with over $25,000 in income is about 2% per year. We are interested in the expected number of audits a person with that income has in a 20-year period. Assume each year is independent.
a. In words, define the random variable X.
b. List the values that X may take on.
c. Give the distribution of X. X ~ _____(_____,_____)
d. How many audits are expected in a 20-year period?
e. Find the probability that a person is not audited at all.
f. Find the probability that a person is audited more than twice
12. The literacy rate for a nation measures the proportion of people age 15 and over that can read and write. The literacy rate in Afghanistan is 28.1%. Suppose you choose 15 people in Afghanistan at random. Let X = the number of people who are literate.
a. Sketch a graph of the probability distribution of X.
b. Using the formulas, calculate the (i) mean and (ii) standard deviation of X.
c. Find the probability that more than five people in the sample are literate. Is it is more likely that three people or four people are literate.
13. Births are approximately uniformly distributed between the 52 weeks of the year. They can be said to follow a uniform distribution from one to 53 (spread of 52 weeks).
a. Graph the probability distribution.
b. ? = _________
c. ? = _________
d. Find the probability that a person is born at the exact moment week 19 starts. That is, find P(x = 19)
e. P(2 < x < 31) = _________
f. Find the probability that a person is born after week 40.
g. P(12 < x|x < 28) = _________
h. Find the 70th percentile.
i. Find the minimum for the upper quarter.
14. A random number generator picks a number from one to nine in a uniform manner. a. Graph the probability distribution.
b. ? = _________
c. ? = _________
d. P(3.5 < x < 7.25) = _________
e. P(x > 5.67)
f. Find the 90th percentile.
15. A subway train on the Red Line arrives every eight minutes during rush hour. We are interested in the length of time a commuter must wait for a train to arrive. The time follows a uniform distribution.
a. Define the random variable. X = _______
b. X ~ _______
c. Graph the probability distribution.
d. f(x) = _______
e. ? = _______
f. ? = _______
g. Find the probability that the commuter waits less than one minute.
h. Find the probability that the commuter waits between three and four minutes. i. Sixty percent of commuters wait more than how long for the train? State this in a probability question, similarly to parts g and h, draw the picture, and find the probability.
16. Suppose that the longevity of a light bulb is exponential with a mean lifetime of eight years.
a. Find the probability that a light bulb lasts less than one year.
b. Find the probability that a light bulb lasts between six and ten years.
c. Seventy percent of all light bulbs last at least how long?
17. The cost of all maintenance for a car during its first year is approximately exponentially distributed with a mean of $150.
a. Define the random variable. X = _________________________________.
b. X ~ = ________
c. ? = ________
d. ? = ________
e. Draw a graph of the probability distribution. Label the axes.
f. Find the probability that a car required over $300 for maintenance during its first year.
18. Suppose that the useful life of a particular car battery, measured in months, decays with parameter 0.025. We are interested in the life of the battery.
a. Define the random variable. X = _________________________________.
b. Is X continuous or discrete?
c. X ~ ________
d. On average, how long would you expect one car battery to last?
e. On average, how long would you expect nine car batteries to last, if they are used one after another?
f. Find the probability that a car battery lasts more than 36 months.
g. Seventy percent of the batteries last at least how long?
19. A web site experiences traffic during normal working hours at a rate of 12 visits per hour. Assume that the duration between visits has the exponential distribution.
a. Find the probability that the duration between two successive visits to the web site is more than ten minutes.
b. The top 25% of durations between visits are at least how long?
c. Suppose that 20 minutes have passed since the last visit to the web site. What is the probability that the next visit will occur within the next 5 minutes?
d. Find the probability that less than 7 visits occur within a one-hour period.
c) 5 | 5 | 0 | 0 P| X|1|2|3|4 0 0-0-0Explanation / Answer
Chapter 4 & 5 – Computation Practice
1. A baker is deciding how many batc
Solution1:
Random variable X is
batches of muffins to make to sell in his bakery.
b. What is the probability the baker will sell more than one batch? P(x > 1) = _
P(X>1)=P(X=2)+P(X=3)+P(X=4)
=0.35+0.4+0.1
=0.85
ANSWER:0.85
c. What is the probability the baker will sell exactly one batch? P(x = 1)
P(x=1)=0.15
On average, how many batches should the baker make?
MEAN=xP(X=x)
=1*0.15+2*0.35+3*0.40+4*0.10
=2.45
On average 2.45 batches baker should make.
Related Questions
drjack9650@gmail.com
Navigate
Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.