Questions and Answers of Statistical Techniques in Business

Let X1, X2 be iid with common distribution having the pdf f(x) = e−x, 0 < x < ∞, zero elsewhere. Show that Z = X1/X2 has an F-distribution.
Using the computer, obtain plots of beta pdfs for α = 1, 5, 10 and β = 1, 2, 5, 10, 20.
Let X have a binomial distribution with parameters n and p = 1/3. Determine the smallest integer n can be such that P(X ≥ 1) ≥ 0.85.
If X is N(1, 4), compute the probability P(1< X2 < 9).
Let X have a Poisson distribution with parameter m. If m is an experimental value of a random variable having a gamma distribution with α = 2 and β = 1, compute P(X = 0, 1, 2).
Suppose X is distributed N2(μ,Σ). Determine the distribution of the random vector (X1+X2,X1−X2). Show that X1+X2 and X1−X2 are independent if Var(X1) = Var(X2).
Let X1 and X2 have a trinomial distribution. Differentiate the moment generating function to show that their covariance is −np1p2.
If a fair coin is tossed at random five independent times, find the conditional probability of five heads given that there are at least four heads.
Show, for k = 1, 2, . . ., n, thatThis demonstrates the relationship between the cdfs of the β and binomial distributions. S P k-1 1²²-¹ (1-2)"-k dz = [ (") p² (1 − p)ª-². T=0 n! (k-1)!(n-k)!
Determine the constant c in each of the following so that each f(x) is a β pdf:(a) f(x) = cx(1 − x)3, 0 < x < 1, zero elsewhere.(b) f(x) = cx4(1 − x)5, 0 < x < 1, zero elsewhere.(c)
Let an unbiased die be cast at random seven independent times. Compute the conditional probability that each side appears at least once given that side 1 appears exactly twice.
Let the random variable X have a distribution that is N(μ, σ2).(a) Does the random variable Y = X2 also have a normal distribution?(b) Would the random variable Y = aX + b, a and b nonzero
Determine the constant c so that f(x) = cx(3 − x)4, 0 < x < 3, zero elsewhere, is a pdf.
Let X equal the number of independent tosses of a fair coin that are required to observe heads on consecutive tosses. Let un equal the nth Fibonacci number, where u1 = u2 = 1 and un = un−1 +
Let the random variable X be N(μ, σ2). What would this distribution be if σ2 = 0?
Let Y have a truncated distribution with pdf g(y) = φ(y)/[Φ(b)−Φ(a)], for a < y < b, zero elsewhere, where φ(x) and Φ(x) are, respectively, the pdf and distribution function of a
Let X1 and X2 be independent random variables. Let X1 and Y = X1+X2 have chi-square distributions with r1 and r degrees of freedom, respectively. Here r1 < r. Show that X2 has a chi-square
Suppose X is a random variable with the pdf f(x) which is symmetric about 0; i.e., f(−x) = f(x). Show that F(−x) = 1 − F(x), for all x in the support of X.
Consider a standard deck of 52 cards. Let X equal the number of aces in a sample of size 2.(a) If the sampling is with replacement, obtain the pmf of X.(b) If the sampling is without replacement,
Assuming a computer is available, plot the pdfs of the random variables defined in parts (a)–(d) of the last exercise. Obtain an overlay plot of all four pdfs also. In R the domain values of the
Consider a shipment of 1000 items into a factory. Suppose the factory can tolerate about 5% defective items. Let X be the number of defective items in a sample without replacement of size n = 10.
A certain job is completed in three steps in series. The means and standard deviations for the steps are (in minutes)Assuming independent steps and normal distributions, compute the probability that
Compute P(X1 + 2X2 − 2X3 > 7) if X1,X2,X3 are iid with common distribution N(1, 4).
Twenty motors were put on test under a high-temperature setting. The lifetimes in hours of the motors under these conditions are given below. Suppose we assume that the lifetime of a motor under
Let X be N(0,1). Use the moment generating function technique to show that Y = X2 is χ2(1).
Suppose the number of customers X that enter a store between the hours 9:00 a.m. and 10:00 a.m. follows a Poisson distribution with parameter θ. Suppose a random sample of the number of customers
Let X, Y , and Z have the joint pdfwhere −∞ < x < ∞, −∞ < y < ∞, and −∞ < z < ∞. While X, Y , and Z are obviously dependent, show that X, Y , and Z are pairwise
The monthly linear trend equation for the Hoopes ABC Beverage Store is:ŷ = 5.50 + 1.25tThe equation is based on 4 years of monthly data and is reported in thousands of dollars. The index for January
Listed below are the price and quantity of several golf items purchased by members of the men’s golf league at Indigo Creek Golf and Tennis Club for 2012 and 2016.a. Determine the simple aggregate
Listed below are the sales at Roberta’s Ice Cream Stand for the last 5 years, 2012 through 2016.Year....................................................
Based on five years of monthly data (the period from January 2011 to December 2015), the trend equation for a small company is ŷ = 3.5 + 0.7t. The seasonal index for January is 120 and for June it
According to a study in Health Magazine, one in three children in the United States is obese or overweight. A health practitioner in Louisiana sampled 500 children and found 210 who were obese or
An instructor has three sections of basic statistics: 8:00 a.m., 10:00 a.m., and 1:30 p.m. Listed ?below are the grades on the first exam for each section. Assume that the distributions do not follow
A book publisher wants to investigate the type of book selected for recreational reading by men and women. A random sample of 540 men and 500 women provided the following information regarding their
A recent census report indicated that 65% of families have two parents present, 20% have only ?mother present, 10% have only a father present, and 5% have no parent present. A random sample of 200
Given the following regression analysis output.a. What is the sample size?b. How many independent variables are in the study?c. Determine the coefficient of determination.d. Conduct a global test of
Given the following regression analysis output:a. What is the sample size?b. Write out the regression equation. Interpret the slope and intercept values.c. If the value of the independent variable is
The results of a one-way ANOVA are reported below. Use the .05 significance level.a. How many treatments are in the study?b. What is the total sample size?c. What is the critical value of F?d. Write
Is there a difference in the mean miles traveled per week by each of two taxicab companies operating in the Grand Strand area? The Sun News, the local paper, is investigating and obtained the
The Philadelphia Area Transit Authority wishes to estimate the proportion of central city workers that use public transportation to get to work. A sample of 100 workers revealed that 64 used public
A recent study of 26 Conway, South Carolina, residents revealed they had lived at their current address an average of 9.3 years. The standard deviation of the sample was 2 years.a. What is the
Americans spend an average (mean) of 12.2 minutes (per day) in the shower. The distribution of times follows the normal distribution with a population standard deviation of 2.3 minutes. What is the
The auditor of the state of Ohio needs an estimate of the proportion of residents who regularly play the state lottery. Historically, about 40% regularly play, but the auditor would like some current
Historically, the proportion of adults over the age of 24 who smoke has been .30. In recent years, much information has been published and aired on radio and TV that smoking is not good for one’s
In recent times, the percent of buyers purchasing a new vehicle via the Internet has been large enough that local automobile dealers are concerned about its impact on their business. The information
The manager of a home improvement store wishes to estimate the mean amount of money spent in the store. The estimate is to be within $4.00 with a 95% level of confidence. The manager does not know
A manufacturer of cell phone batteries wants to estimate the useful life of its battery (in thousands of hours). The estimate is to be within 0.10 (100 hours). Assume a 95% level of confidence and
John Kleman is the host of KXYZ Radio 55 AM drive-time news in Denver. During his morning program, John asks listeners to call in and discuss current local and national news. This morning, John was
Defiance Tool and Die has 293 sales offices throughout the world. The VP of Sales is studying the usage of its copy machines. A random sample of six of the sales offices revealed the following number
The manager of the local Hamburger Express wishes to estimate the mean time customers spend at the drive-through window. A sample of 20 customers experienced a mean waiting time of 2.65 minutes, with
The owner of the Gulf Stream Café wished to estimate the mean number of lunch customers per day. A sample of 40 days revealed a mean of 160 per day, with a standard deviation of 20 per day. Develop
The manager of Tee Shirt Emporium reports that the mean number of shirts sold per week is 1,210, with a standard deviation of 325. The distribution of sales follows the normal distribution. What is
A recent study indicated that women took an average of 8.6 weeks of unpaid leave from their jobs after the birth of a child. Assume that this distribution follows the normal probability distribution
Consider the digits in the phone numbers on a randomly selected page of your local phone book a population. Make a frequency table of the final digit of 30 randomly selected phone numbers. For
Consider all of the coins (pennies, nickels, quarters, etc.) in your pocket or purse as a population. Make a frequency table beginning with the current year and counting backward to record the ages
The IRS offers taxpayers the choice of allowing the IRS to compute the amount of their tax refund. During the busy filing season, the number of returns received at the Springfield Service Center that
For the year 2017, Fred Friendly completed a total of 80 returns. He developed the following table summarizing the relationship between number of dependents and whether or not the client received a
In a memo to all those involved in tax preparation, the IRS indicated that the mean amount of refund was $1,600 with a standard deviation of $850. Assume the distribution of the amounts returned
Fred decided to study the number of exemptions claimed on personal tax returns he prepared in 2017. The data are summarized in the following table.Exemptions
Fred works in a tax office with five other CPAs. There are five parking spots beside the office. In ow many different ways can the cars belonging to the CPAs be arranged in the five spots?
The IRS reports that 15% of returns where the adjusted gross income is more than $1,000,000 will be subject to a computer audit. For the year 2017, Fred Friendly, CPA, completed 16 returns where the
Fred Friendly, CPA, has 20 tax returns to prepare before the April 15th deadline. It is late at night so he decides to do two more before going home. In his stack of accounts, 12 are personal, 5 are
During the 50th Super Bowl, 30 million pounds of snack food were eaten. The chart below depicts this information. a. What is the name given to this graph?b. Estimate, in millions of pounds, the
An investor purchased 200 shares of the Blair Company for $36 each in July of 2013, 300 shares at $40 each in September 2015, and 500 shares at $50 each in January 2016. What is the investor’s
A sample of eight college students revealed they owned the following number of CDs.52 76 64 79 80
The Russell 2000 index of stock prices increased by the following amounts over the last 3 years.18% 4% 2%What is the geometric mean increase for the 3 years?
Listed below is the 2014 median household income for the 50 states and the District of Columbia.https://www.census.gov/hhes/www/income/data/historical/household/Use a statistical software package
Using data from such publications as the Statistical Abstract of the United States, Forbes, or any news source, give examples of variables measured with nominal, ordinal, interval, and ratio scales.
On the Web, go to your favorite news source and find examples of each type of variable. Write a brief memo that lists the variables and describes them in terms of qualitative or quantitative,
Refer to the Lincolnville School District bus data. Prepare a report on the maintenance cost for last month. Be sure to answer the following questions in your report.a. Around what values do the data
Two coins are tossed. If A is the event “two heads” and B is the event “two tails,” are A and B mutually exclusive? Are they complements?
Compute a simple aggregate price index. Use 1990 as the base period. Price 1990 $0.287 Quantity 2016 Item Aluminum (cents per pound) Natural gas (1,000 cu. ft.) Petroleum (barrel) Platinum (troy
Using the period 2000€“02 as the base period, compute a simple index of domestic sales for each year from 2004 to 2015. Domestic Sales Internatlonal Sales Employees (thousands) ($ mllon) ($
Refer to the Baseball 2016 data, which report information on the 2016 Major League Baseball season.a. Rank the teams by the number of wins and their total team salary. Compute the coefficient of rank
Refer to the Baseball 2016 data, which report information on the 30 Major League Baseball teams for the 2016 season. Set up a variable that divides the teams into two groups, those that had a winning
The position of chief of police in the city of Corry, Pennsylvania, is vacant. A search committee of Corry residents is charged with the responsibility of recommending a new chief to the city
Refer to the Baseball 2016 data that report information on the 30 Major League Baseball teams for the 2016 season.a. In the data set, the year opened, is the first year of operation for that stadium.
There are four McBurger restaurants in the Columbus, Georgia, area. The numbers of burgers sold at the respective restaurants for each of the last 6 weeks are shown below. At the .05 significance
The following is a partial ANOVA table.Complete the table and answer the following questions. Use the .05 significance level.a. How many treatments are there?b. What is the total sample size?c. What
A vending machine company sells its packaged foods in a variety of different machines. The company is considering three types of new vending machines. Management wants to know if the different
What is the critical F value when the sample size for the numerator is six and the sample size for the denominator is four?
What is the critical F value when the sample size for the numerator is four and the sample size for the denominator is seven? Use a one-tailed test and the .01 significance level.
The following data was collected for a two-factor ANOVA with three treatments and three blocks.Conduct a test of hypothesis to determine whether the block or the treatment means differ. Using the .05
Compute Paasche€™s index for 2016 using 2000 as the base period. 2016 2000 Price Price $0.81 0.84 1.44 2.91 Item Quantity 18 5 70 27 Quantity 27 Margarine (pound) Shortening (pound) MIlk
Determine a value index for 2016 using 2000 as the base period. 2016 2000 Price Price $0.81 0.84 1.44 2.91 Item Quantity 18 5 70 27 Quantity 27 Margarine (pound) Shortening (pound) MIlk (V2 gallon)
Compute a simple aggregate price index for 2016. Use 2000 as the base period. Price Quantity Part 2016 2000 2000 2016 340 130 250 RC-33 SM-14 320 110 230 $0.50 1.20 0.85 $0.60 0.90 1.00 WC50
Compute Paasche€™s index for 2016 using 2000 as the base period. Price Quantity Part 2016 2000 2000 2016 340 130 250 RC-33 SM-14 320 110 230 $0.50 1.20 0.85 $0.60 0.90 1.00 WC50
Compute Laspeyres€™ price index for 2016 using 2000 as the base period. Price Quantity 2016 2000 2016 $0.05 2000 Item Cabbage (pound) Carrots (bunch) Peas (quart) Endive (bunch) 1,500 200
Compute Paasche€™s index for 2016 using 2000 as the base period. Price 2016 $0.05 Quantity 2000 2,000 200 400 100 2000 2016 Item 1,500 200 500 200 Cabbage (pound) Carrots (bunch) Peas
Determine Fisher€™s ideal index using the values for the Laspeyres and Paasche indexes computed in the two previous problems. Price 2016 $0.05 Quantity 2000 2,000 200 400 100 2000 2016 Item
Determine a value index for 2016 using 2000 as the base period. Price 2016 $0.05 Quantity 2000 2,000 200 400 100 2000 2016 Item 1,500 200 500 200 Cabbage (pound) Carrots (bunch) Peas (quart) Endive
Compute Laspeyres€™ price index for 2016 using 1990 as the base period. Price 1990 $0.287 Quantity 2016 Item Aluminum (cents per pound) Natural gas (1,000 cu. ft.) Petroleum (barrel)
Determine Fisher€™s ideal index using the values for the Laspeyres and Paasche indexes computed in the two previous problems. 2000 2016 Item Price Price Quantity 18 Quantity 27 Margarine
Compute Paasche's index for 2016 using 1990 as the base period. 2000 2016 Price $0.81 0.84 1.44 2.91 Item Price Quantity 18 Quantity 27 Margarine (pound) Shortening (pound) Milk (2 gallon) Potato
Team Sports Inc. sells sporting goods to high schools and colleges via a nationally distributed catalog. Management at Team Sports estimates it will sell 2,000 Wilson Model A2000 catcher’s mitts
Refer to Exercises 1, 3, and 5. Compute the expected value of perfect information. State of Nature Alternative s, $50 $70 $100 A, Az A, 90 40 80 90 70 60
Refer to Exercises 2, 4, and 6. Compute the expected value of perfect information. State of Nature Alternative s, $50 $70 $100 A, Az A, 90 40 80 90 70 60
Dude Ranches Incorporated was founded on the idea that many families in the eastern and southern areas of the United States do not have a sufficient amount of vacation time to drive to the dude
Tim Waltzer owns and operates Waltzer€™s Wrecks, a discount car rental agency near Cleveland Hopkins International Airport. He rents a wreck for $20 a day. He has an arrangement with
Compute a simple index for the number of employees for GE using the period 2004€“06 as the base. What can you conclude about the change in the number of employees over the period? Employees

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