QM Exam 2
(1 - POISSON(3, 3, 1)) x POISSON(1, 3, 1) x 2
Assume calls arrive at the switchboard following a Poisson distribution with mean μ = 3 calls per minute. What is the probability that during a two-minute period we get more than three calls in one of the minutes and fewer than two calls in the other?
.5 ± .0098, no, since n=100, CLT is invoked n = 385
A machine produces rolls of tape of any desired width (and length). It is known that the machine's capability is such that the standard deviation of the width of an individual roll is .05 inches, . If a random sample of 100 rolls of tape indicates a sample mean of .5 inches find a 95% confidence interval for the true mean of the order of (over 1,000,000) rolls. Do we need to assume that the distribution of widths is normally distributed? If we wanted to estimate the mean of the process in question above within .005 inches with 95% confidence, what sample size of rolls would we need? How can you most easily determine whether it is more than the n = 100 ?
.832 ---- .908
A random sample of women business executives were asked whether they prefer "Ms." as a title, compared to either "Mrs." or "Miss." Out of 300 women polled, 261 said that they preferred "Ms." Find a 95% confidence interval for the true proportion of women business executives that prefer "Ms."
18.2 ± 2.60
A randomly selected 25 SUV owners in the United States was asked to report the (highway) miles per gallon of their SUV. The results revealed a mean of 18.2, with a [sample] standard deviation of 6.3. Find a 95% confidence interval for the true miles per gallon for SUVs.
180 ± 14.04
A recent study by the New England Auto Dealers Association revealed that the mean profit per car sold for a random sample of 20 cars was $180 with a standard deviation of $30. Find a 95% confidence interval for the true average profit per car sold.
$538,833 ± $290,778
Annalise Realty would like to develop a 99% confidence interval for the true average price of homes ( μ ) in Wrentham, Ma. She randomly selected 6 homes from Wrentham and found the following values: $237,000 $436,000 $550,000 $640,000 $660,000 $710,000 Please use Excel to find a 99% confidence interval for the true average price of homes in Wrentham.
a) 42 b) 42 ± 3.92 c) 42, 42 ± 2.94, same confidence, increased precision
A research firm conducted a survey to determine the mean amount of time teenagers spend each day texting. The standard deviation is known to be 12 minutes. A sample of 36 teenagers revealed a sample mean of 42 minutes. a) What is a point estimate of the population mean? b) Find a 95% confidence interval estimate for μ? c) Repeat the exercise above, this time with a sample of 64 teenagers.What now is the 95% confidence interval estimate for μ? Compare your answer with the previous problem. Which interval has more confidence? Which interval has more precision?
P(getting at least a "B") = 1 - BINOMDIST(7, 10, .8, 1) = .678 P(Passing the exam) = 1 - BINOMDIST(5, 10, .8, 1) = .967 P(getting at least a "B") / P(passed the exam) = .678/.967 = .701
A teacher gave her students a multiple-choice exam consisting of ten questions, each question having five multiple-choice answers. Fiona, a student in the class, has determined that she needs eight out of ten correct to receive a grade of at least "B" for the course. Fiona has also determined that she has an 80% chance of getting any one question correct, and each question is independent of each other question. When Fiona saw her teacher later that day in the hallway she asked her teacher what she scored on the exam. The teacher replied that she does not remember the exact score, but she does remember that she definitely passed the exam. If a passing grade is at least 6 correct, what is the probability that Fiona will get at least a "B" for the course?
a) 1 - normdist( 65, 50, 10, 1) or .0668 b) normdist( 40, 50, 10, 1) or .1587 c) normdist( 55, 50, 10, 1 ) - normdist( 45, 50, 10, 1 ) or .3830 d) Norminv ( .90, 50, 10 ) or 62.82 e) Norminv( .95, 50, 10 ) or 66.45
Assume that the length of time needed to renew your license at the Massachusetts RMV is normally distributed with mean μ = 50 minutes, and a standard deviation σ = 10 minutes. a. What is the probability that a randomly selected person will require more than 65 minutes to renew their license? b. What is the probability that a randomly selected person will require less than 40 minutes. c. What is the probability that a randomly selected person will require between 45 and 55 minutes to renew their license? d. Assume that the length of time needed to renew your license at the Massachusetts RMV is normally distributed with mean μ = 50 minutes, and a standard deviation σ = 10 minutes. If the RMV wishes to give a discount to those 10% of customers who wait the longest, what is the cutoff to get the discount? e. The RMV wishes to launch a new public relations campaign. Their slogan is "95% of all customers will have their licenses renewed in under ______ minutes. Fill in the blank?
a) P( less than 178 ) = NORMDIST( 178, 160, 15, 1 ) = 0.88493 - P( less than 145 ) = NORMDIST( 145, 160, 15, 1 ) = 0.15865 0.72628 0.72628 x 20,000 = 14,526 rounded b) 0.69462
Assume that the weight of American males is normally distributed with a mean of 160 pounds, with a standard deviation of 15 pounds. As a store manager, you wish to order 20,000 (male) bathrobes. These bathrobes come in three sizes, Small, Medium, and Large. The weight/size table for male bathrobes is: Male weight Bathrobe size Under 145 pounds Small 145 - 178 pounds Medium 178 pounds or more Large a) If you wish to order the same proportion of sizes as in the male population, how many medium size bathrobes should you order? b) If two men enter the store and each buys a bathrobe, what is the probability that the less heavy of the two men buys a medium?
98%
For a sample size of n = 20, with s = 5, we have constructed the following confidence interval for our estimate of the mean of a normal distribution: 7.161 to 12.839. What confidence level did we use?
i) tinv(.05,9) ii) tinv(.01,24) iii) tinv(.05,30) iv) tinv(.10,9) b) $6,780 +/- 410.70 c) 5 ± 4.598 d) 5 ± 1.48 e) standard deviations are different f) yes, n<30
For the following values of the confidence level and sample size, find the appropriate t-table value: i) 1-α = .95, n = 10 ii) 1-α = .99, n = 25 iii) 1-α = .95, n = 31 iv) 1-α = .90, n = 10 b) A bank manager wants to estimate the average balance, of a certain class of savings accounts. He samples a random selection of 30 accounts, and finds that X-bar = $6,780, and s = $1,100. Find a 95% confidence interval for the (true) average balance of all of the thousands of savings accounts at the bank of that certain class. c) Find a 95% confidence interval for the true mean of the normally distribbuted population from which a random sample of six is: (1,1,1,9,9,9) d) Repeat part c for data: (3,4,5,5,6,7) e) Parts c and d have the same center, so why are the confidence intervals so different. f) Was the fact that the population probability distribution was stated to be normal necessary for you to solve parts c) and d)?
n = 2401; n = 2017
If we wanted to estimate the true proportion of women business executives who preferred the "Ms." title within .02 with 95% confidence, what sample size of women business executives is needed? How would the answer change if we were certain that the true proportion who prefer the "Ms." title is no lower than 0.7?
200 ± 11.598
In a production process, the average daily output per worker is needed in order to establish pay scales. It is assumed that output is normally distributed. A random sample of 20 workers was taken and X-bar = 200, and s = 30. Find a 90% confidence interval for the (true) mean output for the entire population of workers
a) binomdist(12,12,.7,0) b) binomdist(9,12,.7,0) c) 1-binomdist(5,12,.7,1) d) 1-binomdist(8,12,.7,1) e) (1-binomdist(5,12,.7,1))^2
In a recent study, 70% of homes in the United States were found to have a laptop computer. In a random sample of 12 homes, what is the probability that: a) All 12 have a laptop b) Only 9 have a laptop c) at least 6 have a laptop? d) more than 8 have a laptop? e) if you randomly sample 12 homes today and 12 homes tomorrow what is the probability that you will find at least 6 haivng laptops each day?
323
Jane Isaacson wishes to estimate p, the true proportion of college students who use a computer at least an hour per week. She wants to be 95% confident that the estimate she obtains is within 0.05 of the true value. What minimum sample size does Jane require? She knows from past studies that p is between .7 and .95)
Norminv( .95, 100, 20) or 132.9 ≈ 133
Nicole has determined that the average monthly sales of rabbits at "Nicole's Rabbit World" are normally distributed with a mean of 100 and a standard deviation of 20. Nicole wants to set an inventory level such that there is only a 5% chance of running out of stock. Where should Nicole set the inventory level?
1 - BINOMDIST(13, 15, Binomdist(1,20,.02,1), 1)
Noah Michael tests and inspects several batches of product per month. Each batch has 1000 items in it, and each item is classified as either defective or good. The inspection of a batch consists of taking 20 randomly-selected products, with replacement, and determining how many of the 20 products are defective. A batch passes inspection if at most one of the 20 products are defective. If Sam inspects 15 batches in a given month, and if the probability that any given product is defective is .02, what is the probability that at least 14 of the 15 batches pass inspection?
a) binomdist(10,10,.75,0) b) binomdist(6,10,.75,0) c) binomdist(4,10,.75,1) d) 1-binomdist(7,10,.75,1) e) 1-binomdist(7,10,.75,1) f) binomdist(4,10,.75,1)
Noel plays basketball for the King Philip Metrowest Basketball Team. When a player in basketball gets fouled he is often awarded free throw attempts. Assume that Noel makes 75% of his free throws and that each free throw attempt is independent of any prior free throw attempt Noel has taken. If Noel takes 10 free throw attempts in a game, what is the probability he makes a) All 10 b) Exactly 6? c) Less than 5 d) more than 7) e) at least 8? f) at most 4?
a) Poisson(10,12,0) b) Poisson(12,12,0) c) binomdist(3,4,1-poisson(10,12,1),0)
Scoring for the King Phillip 6th grade basketball team follows a Poisson distribution with a rate of 1.5 points scored per minute and each quarter in the 6th grade league last 8 minutes . a) What is the probability of the team scoring exactly 10 points in a quarter? b) Scoring 12 points in a given quarter? c) If you assume that each quarter in a game is independent of any other, what is the probability of the King Philip team scoring over 10 points for exactly 3 out of 4 quarters of a game?
Can only be found using excel. 95.44% Confidence
Sofia has accumulated waiting time data from the Wrentham and Norfolk offices of the Metrowest Dental Associates. (Sofia knows that waiting time per patient at each office is normally distributed with known standard deviation as shown) sample size: wrentham = 25; norfolk = 16 sample mean: wrentham = 16; norfolk = 20 standard deviations: wrentham = 5; norfolk = 12 At what confidence level will the Wrentham and Norfolk lower confidence limits be equal?
binomdist(199, 400, .55, 1) or 1 - binomdist(200, 400, .45, 1)
Suppose that 55% of all people prefer Coca Cola to Pepsi Cola. If we randomly sample 400 people, what is the probability that the majority of respondents indicate a preference for Pepsi Cola? Assume that each person's opinion is independent from each other person's opinion.
(1-Normdist( 700,500,80,1 ))/(1-Normdist( 540,500,80,1 ))
Suppose that GMAT scores of applicants to a certain business school, ABC, are normally distributed with mean μ = 500 and standard deviation σ = 80. Suppose further that ABC automatically accepts any student who's GMAT score exceeds 700. The executive committee of the graduate admissions department is given only those applications with GMAT score over 540 (the rest are put into a "desperate-only" file). What proportion of the applications received by the executive committee gets automatic acceptance?
1 - binomdist( 0, 4, 1-normdist( 120, 100, 10, 1), 1)
Suppose that X, the length of a roll of yarn, is normally distributed with mean μ = 100 feet and standard deviation σ = 10 feet. If we take a random sample of four rolls of yarn, what is the probability that the longest of the four rolls has a length over 120 feet?
a) binomdist(11, 100, .1, 0) b) 1 - binomdist(10, 100, .1, 1) c) {1 - binomdist(10, 100, .1, 1)}/ {1 - binomdist(9, 100, .1, 1)} d) 1
Suppose that a certain target market has a 10% response rate to a catalog sent out by a well-known clothing store. Suppose that a special audit is done on 100 randomly selected recipients of the catalog. a) What is the probability that among the 100 there are exactly 11 responses? b) What is the probability that among the 100 there are at least 11 responses? c) What is the probability that among the 100 there are at least 11 responses, given we know we got at least the expected number of responses? d) If we know that the company received only 1 response from the first 90 recipients audited, what is the expected number of responses from the last 10 recipients about to be audited?
1 - BINOMDIST(48, 50, 1 - BINOMDIST(97, 100, .99, 1), 1)
Suppose that at American Razor Corporation (ARC), random sets of 100 razor blades are tested. A set is deemed acceptable if at least 98 of the 100 pass a sharpness test. The probability of any one blade passing this test is .99, and each blade is considered independent of each other blade. If ARC tests 50 sets of blades, what is the probability that at least 49 of the 50 sets are acceptable?
1-binomdist( 19, 50, 1-normdist(300,260,35,1), 1)
Suppose that the average weight of football players is normally distributed with a mean μ = 260 lbs, and a standard deviation σ = 35 lbs. If we randomly sample 50 players, what is the probability that at least 20 weigh more than 300lbs?
60. C
Suppose that the distribution of weights of a large batch of items is normally distributed with a mean μ and standard deviation σ. If we take a sample size of 100, the answer to the question, "What is the probability that XBar is greater than 1.5μ?" a) depends on the value of μ, but not the value of σ b) depends on the value of σ, but not the value of μ c) depends on the value of both μ and σ d) does not depend on the value of μ, nor on the value of σ e) depends on how many home runs David (Papi) Ortiz hits.
Poisson(8, 6, 1)
Suppose that the number of cars entering a car wash on Thursday mornings in the summertime is Poisson distributed with a mean of μ = 6/hour. If the car wash opens at 9:00 a.m., what is the probability that the 9th car to arrive at the car wash on a random Thursday morning in the summer arrives after 10:00 a.m.?
a) 1 - Poisson( 10, 7, 1 ) b) AFB has fewer than 10 lightning strikes? X ≤ 9 Poisson( 9, 7, 1 ) c) No. it omits X = 10 d)
Suppose that the number of lightning strikes per year at the AFB Golf course in Boca Raton, Florida, is Poisson distributed with a mean of 7. a) What is the probability that a given year at AFB has more than 10 lightning strikes? b) What is the probability that a given year at c) Do the answers to a) and b) add to 1? If not, why not? d) Starting with next year, what is the probability that AFB encounters a year of above average number of lightning strikes before it encounters a year of below average number of lightning strikes?
BINOMDIST(5, 5, POISSON( 2, 3, 1 ), 0)
Suppose that the number of phone calls that a business receives is Poisson distributed with mean, μ = 6/hour. What is the probability that the business has five straight half-hour periods in which each half-hour has no more than two calls?
{1 - NORMDIST(200, 175, 25, 1)} / .5
Suppose that the weight of men in the United States is normally distributed with μ = 175 pounds and σ = 25 pounds. What is the probability of a randomly-selected man weighing more than 200 pounds given that we know his weight is above average?
a) W b) W c) W d) ?
Suppose that we have sampled n observations from a normal distribution with known standard deviation, σ, and found a 95% confidence interval for μ. For each of the following four "changes," indicate whether the interval will definitely get wider (W), definitely get narrower (N), or will not be definite either way (?). Each of the four changes are to be assumed to be totally independent questions, and in each case, everything except what is mentioned as changed is assumed to stay the same. Circle one choice per part: a) Increased confidence level (W, N, ?) b) decreased sample size and increased SD w,n,? c) increased pop. size from relatively small to relatively large d) increased sample size and increased confidence level
44
Suppose that μ is the mean of a normal distribution. What sample size is required such that the probability is 90% that the sample mean is within a quarter of a standard deviation of μ?
11
Suppose we want to estimate the average weight of men in the United States within 10 pounds. σ is known to be 20 pounds. If we want to be 90% confident in our estimate, how large a sample size is required?
.24 ± .059
Suppose you wish to estimate the proportion of men in the United States who weigh over 200 pounds. You randomly sample 200 U.S. men and find that 48 weigh over 200 pounds. Find a 95% confidence interval for the proportion of U.S. men who weigh over 200 pounds.
a) Poisson(0,10,0) b) 1-Poisson(2,10,1) c) Poisson(5,10,1)
Taylor, a statistics professor, holds office hours every Thursday. Students arrive at her office at the rate of 10 per hour. The number of arrivals follows a Poisson distribution. a) What is the probability that no students will arrive in a particular hour? b) What is the probability that at least 3 arrive in a particular hour? c) What is the probability that at most 5 will arrive in a particular hour?
µ = 45.09, σ = 11.65
The daily amount of time that teenagers spend playing XBOX follows a normal distribution. However, the mean and the standard deviation are unknown. Forty percent of the time, playing time exceeds 48 minutes, 10 percent of the time, playing time exceeds 60 minutes. What is the mean and standard deviation?
i) .9876 ii) .5 iii) .1056 iv) .0228 v) 11.484 vi) 11.664 vii) no, since n=100, CLT is invoked b) i) .3830 iii) .4013 v) 9.43 vii) yes, n<30, CLT is not invoked
The length of long distance telephone calls ( X ) have a mean, μ = 12 minutes, and standard deviation, σ = 4 minutes. If a random sample of 100 calls is selected, i) What is the probability that the sample mean, x-bar, is between 11 and 13 minutes? ii) What is the probability that the sample mean, x-bar, exceeds 12 minutes? iii) What is the probability that the sample mean, x-bar, exceeds 12.5 minutes? iv) What is the probability that the sample mean, x-bar, is under 11.2 minutes? v) There is a 90% chance that the sample mean, x-bar, will be above how many minutes vi) The lowest 20% of the sample means will be below what number of minutes? vii) To answer the above questions, do you need to assume that the probability distribution of individual long distance phone call lengths is normally distributed? b) Repeat the odd-numbered "sub-parts" of part a) if the sample size is n=4, instead of n=100.
a) 1 - Poisson(20, 24, 1) b) {1 - Poisson(6, 8, 1)}3 = binomdist(3, 3, 1-Poisson(6, 8, 1), 0) c) binomdist(1, 3, 1-Poisson(12, 8, 1), 0)
The number of arrivals at a walk-in medical clinic in the 9am - noon time period on a weekday follows a Poisson distribution with mean, 8/hour. a) What is the probability that there are more than 20 arrivals during a randomly selected 9am - noon weekday time period? b) What is the probability that during a particular 9am - noon weekday time period, each of the three individual hours has at least 7 arrivals? c) What is the probability that during a particular 9am - noon weekday time period, exactly one of the three individual hours has more than 12 arrivals?
18 ± 1.4
The owner of Aleena's Poultry Farm wants to estimate the mean number of eggs laid per chicken. A sample of 20 chickens shows they laid an average of 18 eggs per month with a standard deviation of 3 eggs per month. Find a 95% confidence interval for the population mean. Would it be reasonable to conclude that the population mean is 23 eggs?
.75, .75 ± .11
The owner of the Regional Service Center wishes to estimate the true proportion of their customers who pay with a credit/debit card. He surveyed 100 customers and found that 75 paid with a credit/debit card. Estimate the value of the true population proportion. Find a 99% confidence interval for the true population proportion.
unable to be determined
We have a random variable, X, that is normally distributed and P( X > 5 ) = 0.1. What is P(X > -5)? (Circle your choice of answer) i) .1 ii) .4 iii) .6 iv) .9 v) unable to be determined
.9
We have a random variable, X, that is normally distributed with μX = 0 and P( X > 5 ) = 0.1. What is P(X > -5)? (Circle your choice of answer) i) .1 ii) .4 iii) .6 iv) .9 v) unable to be determined
B
Which of the following one choice is true regarding the probability distribution of XBar, for a sufficiently large sample size? a) It has the same shape as the population distribution, with smaller mean and smaller standard deviation. b) It has a normal distribution, with same mean as the population distribution but with a smaller standard deviation. c) It has the same shape, mean, and standard deviation as the population distribution. d) It has the same shape and mean as the population distribution, with smaller standard deviation. e) It has a normal distribution, with the same mean and standard deviation as the population distribution.
2401
You desire to estimate p for a binomial process. You know that p is less than 0.65 and greater than 0.45. Find n, the sample size required, to yield 95% confidence of estimating the true proportion, p, within .02.
