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Business Analytics Data Analysis And Decision Making 6th Edition S. Christian Albright, Wayne L. Winston - Solutions

If your company makes a particular decision in the face of uncertainty, you estimate that it will either gain $10,000, gain $1000, or lose $5000, with probabilities 0.40, 0.30, and 0.30, respectively. You (correctly) calculate the EMV as $2800. However, you distrust the use of this EMV for
In the previous question, suppose you have the option of receiving a check for $2700 instead of making the risky decision described. Would you make the risky decision, where you could lose $5000, or would you take the sure $2700? What would influence your decision?
In a classic oil-drilling example, you are trying to decide whether to drill for oil on a field that might or might not contain any oil. Before making this decision, you have the option of hiring a geologist to perform some seismic tests and then predict whether there is any oil or not. You assess
Your company has signed a contract with a good customer to ship the customer an order no later than 20 days from now. The contract indicates that the customer will accept the order even if it is late, but instead of paying the full price of $10,000, it will be allowed to pay 10% less, $9000, due to
You must make one of two decisions, each with possible gains and possible losses. One of these decisions is much riskier than the other, having much larger possible gains but also much larger possible losses, and it has a larger EMV than the safer decision. Because you are risk averse and the
A potentially huge hurricane is forming in the Caribbean, and there is some chance that it might make a direct hit on Hilton Head Island, South Carolina, where you are in charge of emergency preparedness. You have made plans for evacuating everyone from the island, but such an evacuation is
It seems obvious that if you can purchase information before making an ultimate decision, this information should generally be worth something, but explain exactly why (and when) it is sometimes worth nothing?
Insurance companies wouldn't exist unless customers were willing to pay the price of the insurance and the insurance companies were making a profit. So explain how insurance is a win-win proposition for customers and the company.
Suppose that you want to know the opinions of American secondary school teachers about establishing a national test for high school graduation. You obtain a list of the members of the National Education Association (the largest teachers' union) and mail a questionnaire to 3000 teachers chosen at
What is the difference between a standard deviation and a standard error? Be precise.
Explain as precisely as possible what it means that the sample mean is an unbiased estimate of the population mean [as indicated in Equation (7.1)]? E(X̅) = μ.
Explain the difference between the standard error formulas in equations (7.2) SE(X̅) = σ / √n and (7.3). Why is Equation (7.3) SE (X̅) = s/√n the one necessarily used in real situations?
Explain as precisely as possible what Equation X̅ ± 2 s/√n (7.4) means, and the reason for the 2 in the formula?
Explain as precisely as possible the role of the finite population correction. In which types of situations is it necessary? Is it necessarily used in the typical polls you see in the news?
In the wheel of fortune simulation with, say, three spins, many people mistakenly believe that the distribution of the average is the flat graph in Figure 7.9, that is, they believe the average of three spins is uniformly distributed between $0 and $1000. Explain intuitively why they are
Explain the difference between a point estimate for the mean and a confidence interval for the mean. Which provides more information?
Explain as precisely as possible what the central limit theorem says about averages?
Many people seem to believe that the central limit theorem "kicks in" only when n is at least 30. Why is this not necessarily true? When is such a large n necessary?
Suppose you are a pollster and are planning to take a sample that is very small relative to the population. In terms of estimating a population mean, can you say that a sample of size 9n is about 3 times as accurate as a sample of size n? Why or why not? Does the answer depend on the population
A sportswriter wants to know how strongly the residents of Indianapolis, Indiana, support the local minor league baseball team, the Indianapolis Indians. He stands outside the stadium before a game and interviews the first 30 people who enter the stadium. Suppose that the newspaper asks you to
You saw in Equation (7.1) E(X̅) = μ. That the sample mean is an unbiased estimate of the population mean. However, some estimates of population parameters are biased. In such cases, there are two sources of error in estimating the population parameter: the bias and the standard error. To
A large corporation has 4520 male and 567 female employees. The organization's equal employment opportunity officer wants to poll the opinions of a random sample of employees. To give adequate attention to the opinions of female employees, exactly how should the EEO officer sample from the given
Suppose that you want to estimate the mean monthly gross income of all households in your local community. You decide to estimate this population parameter by calling 150 randomly selected residents and asking each individual to report the household's monthly income. Assume that you use the local
Provide an example of when you might want to take a stratified random sample instead of a simple random sample, and explain what the advantages of a stratified sample might be?
Provide an example of when you might want to take a cluster random sample instead of a simple random sample, and explain what the advantages of a cluster sample might be. Also, explain how you would choose the cluster sample?
Do you agree with the statement that non-response error can be overcome with larger samples? If you agree, explain why. If you disagree, provide an example that backs up your opinion.
When pollsters take a random sample of about 1000 people to estimate the mean of some quantity over a population of millions of people, how is it possible for them to estimate the accuracy of the sample mean?
Calculate the following probabilities using Excel. (If you have Excel 2010 or later, we suggest using its new functions.)a. P(t10 ≥ 1.75), where t10 has a t distribution with 10degrees of freedom.b. P(t100 ≥ 1.75), where t100 has a t distribution with 100 degrees of freedom. How do you
Calculate the following quantities using Excel. (If you have Excel 2010 or later, we suggest using its new functions.)a. P(−2.00 ≤ t10 ≤ 1.00), where t10 has a t distribution with 10 degrees of freedom.b. P(−2.00 ≤ t100 ≤ 1.00), where t100 has a t distribution with 100 degrees of
Calculate the following quantities using Excel. (If you have Excel 2010 or later, we suggest using its new functions.)a. Find the value of x such that P(t10 > x) = 0.75, where t10 has a t distribution with 10 degrees of freedom.b. Find the value of y such that P(t100 > y) = 0.75, where t100
Under what conditions, if any, is it not correct to assume that the sampling distribution of the sample mean is approximately normally distributed?
Researchers often create multiple 95% confidence intervals based on a given data set. For example, if the variable of interest is home price and there are five neighborhoods in the population, they might create 10 confidence intervals, one for each difference between mean home prices for a given
Based on a given random sample, suppose you calculate a 95% confidence interval for the following difference: the mean test score for students under 25 years old minus the mean test score for students at least 25 years old, and the confidence interval extends from -14.3 to 1.2. How would you
When, if ever, is it appropriate to use the standard normal distribution as a substitute for the t distribution with n − 1 degrees of freedom in estimating a population mean?
"Assuming that all else remains constant, the length of a confidence interval for a population mean increases whenever the confidence level and sample size increase simultaneously." Is this statement true or false? Explain your choice.
"The probability is 0.99 that a 99% confidence interval contains the true value of the relevant population parameter." Is this statement true or false? Explain your choice.
Suppose you have a list of salaries of all professional athletes in a given sport in a given year. For example, you might have the salaries of all Major League Baseball players in 2016. Does it make sense to find a 95% confidence interval for the mean salary? If so, what is the relevant population?
Suppose that someone proposes a new way to calculate a 95% confidence interval for a mean. This could involve any arithmetic on the given data. For example, it could say to go out 1.75 inter quartile ranges (IQRs) on either side of the median. What would it mean to say that this procedure produces
The sample size formula for a confidence interval for the population mean requires an estimate of the population standard deviation. Intuitively, why is this the case? Specifically, why is the required sample size larger if the population standard deviation is larger?
Suppose a 95% confidence interval for a population mean has been calculated, and it extends from 123.7 to 155.2. Some people would then state, "The probability that the population mean is between 123.7 and 155.2 is 0.95." Why is this, strictly speaking, an invalid statement? How would you rephrase
Suppose you are testing the null hypothesis that a mean equals 75 versus a two-tailed alternative. If the true (but unknown) mean is 80, what kind of error might you make? When will you not make this error?
Suppose that you are performing a one-tailed hypothesis test. "Assuming that everything else remains constant, a decrease in the test's level of significance (α) leads to a higher probability of rejecting the null hypothesis." Is this statement true or false? Explain your reasoning?
Can pleasant aromas help people work more efficiently? Researchers conducted an investigation to answer this question. Fifty students worked a paper-and-pencil maze ten times. On five attempts, the students wore a mask with floral scents. On the other five attempts, they wore a mask with no scent.
Suppose you hear the claim that a given test, such as the chi-square test for normality, is not very powerful. What exactly does this mean? If another test, such as the Lilliefors test, is claimed to be more powerful, how is it better than the less powerful test?
Explain exactly what it means for a test statistic to fall in the rejection region?
Give an example of when a one-sided test on a population mean would make more sense than a two tailed test. Give an example of the opposite. In general, why do we say that there is no statistical way to decide whether a test should be run as a one-tailed test or a two-tailed test?
For any given hypothesis test, that is, for any specification of the null and alternative hypotheses, explain why you could make only a type I error or a type II error, but not both. When would you make a type I error? When would you make a type II error? Answer as generally as possible
What are the null and alternative hypotheses in the chi-square or Lilliefors test for normality? Where is the burden of proof? Might you argue that it should go in the other direction? Explain.
We didn't discuss the role of sample size in this chapter as thoroughly as we did for confidence intervals in the previous chapter, but more advanced books do include sample size formulas for hypothesis testing. Consider the situation where you are testing the null hypothesis that a population mean
Suppose that you wish to test a researcher's claim that the mean height in meters of a normally distributed population of rosebushes at a nursery has increased from its commonly accepted value of 1.60. To carry out this test, you obtain a random sample of size 150 from this population. This sample
Suppose that you wish to test a manager's claim that the proportion of defective items generated by a particular production process has decreased from its long-run historical value of 0.30. To carry out this test, you obtain a random sample of 300 items produced through this process. The test
A common complaint of university students is that tenure-track instructors care a lot more about their research than their teaching and general availability to students. The file P10_48.xlsx contains hypothetical data on 75 tenure-track instructors where you can check whether these complaints are
The file P03_55.xlsx lists the average salary for each Major League Baseball (MLB) team from 2004 to 2011, along with the number of team wins in each of these years.a. Rearrange the data so that there are four long columns: Team, Year, Salary, and Wins. There should be 8*30 values for each.b.
As indicated by the opening vignette to this chapter, a motel chain could use regression to determine good locations for new motels from data on existing motels. The data in the file P10_63.xlsx provide one possibility. The Existing Sites sheet contains data on 90 motels in the chain. The cell
Sometimes it is instructive to generate random data that have a given relationship and then use regression on the generated data to see if the estimated regression is basically the same as that used to generate the data in the first place.a. Let X be normally distributed with mean 100 and standard
Consider the relationship between yearly wine consumption (liters of alcohol from drinking wine, per person) and yearly deaths from heart disease (deaths per 100,000 people) in 19 developed countries. Suppose that you read a newspaper article in which the reporter states the following: Researchers
Explain the benefits of using natural logarithms of variables, either of Y or of the X's, as opposed to other possible nonlinear functions, when scatter plots (or possibly economic considerations) indicate that nonlinearities should be taken into account. Explain exactly how you interpret
The number of cars per 1000 people is known for virtually every country in the world. For many countries, however, per capita income is not known. How might you estimate per capita income for countries where it is unknown?
"It is generally appropriate to delete all outliers in a data set that are apparent in a scatter-plot." Do you agree with this statement? Explain.
How would you interpret the relationship between two numeric variables when the estimated least squares regression line for them is essentially horizontal (i.e., flat)?
Suppose that you generate a scatter-plot of residuals versus fitted values of the dependent variable for an estimated regression equation. Furthermore, you find the correlation between the residuals and fitted values to be 0.829. Does this provide a good indication that the estimated regression
Suppose that you have generated three alternative multiple regression equations to explain the variation in a particular dependent variable. The regression output for each equation can be summarized as follows:Which of these equations would you select as "best"? Explain your choice.
Suppose you want to investigate the relationship between a dependent variable Y and two potential explanatory variables X1 and X2. Is the R2 value for the equation with both X variables included necessarily at least as large as the R2 value from each equation with only a single X? Explain why or
Suppose you believe that two variables X and Y are related, but you have no idea which way the causality goes. Does X cause Y or vice versa (or maybe even neither)? Can you tell by regressing Y on X and then regressing X on Y? Explain. Also, provide at least one real example where the direction of
Suppose you have two columns of monthly data, one on advertising expenditures and one on sales. If you use this data set, as is, to regress sales on advertising, will it adequately capture the behavior that advertising in one month doesn't really affect sales in that month but only in future
Suppose you want to predict reading speed using, among other variables, the device the person is reading from. This device could be a regular book, an iPad, a Kindle, or others. Therefore, you create dummy variables for device. How, exactly, would you do it? If you use regular book as the reference
The Durbin-Watson test is for detecting lag 1 autocorrelation in the residuals. Which values of DW signal positive autocorrelation? If you observe such a DW value but ignore it, what might go wrong with predictions based on the regression equation? Specifically, if the data are time series data,
Explain why it is not possible to estimate a linear regression model that contains all dummy variables associated with a particular categorical explanatory variable.
Suppose you have a data set that includes all of the professional athletes in a given sport over a given period of time, such as all NFL football players during the 2014-2016 seasons, and you use regression to estimate a variable of interest. Are the inferences discussed in this chapter relevant?
Distinguish between the test of significance of an individual regression coefficient and the ANOVA test. When, if ever, are these two statistical tests essentially equivalent?
Which of these intervals based on the same estimated regression equation with fixed values of the explanatory variables would be wider: (1) a 95% prediction interval for an individual value of Y or (2) a 95% confidence interval for the mean value of Y? Explain your answer. How do you interpret the
Regression outputs from virtually all statistical packages look the same. In particular, the section on coefficients lists the coefficients, their standard errors, their t-values, their p-values, and (possibly) 95% confidence intervals for them. Explain how all of these are related.
If you are building a regression equation in a forward stepwise manner, that is, by adding one variable at a time, explain why it is useful to monitor the adjusted R2 and the standard error of estimate. Why is it not as useful to monitor R2?
You run a regression with two explanatory variables and notice that the p-value in the ANOVA table is extremely small but the p-values of both explanatory variables are larger than 0.10. What is the probable reason? Can you conclude that neither explanatory variable does a good job in predicting
Why are outliers sometimes called influential observations? What could happen to the slope of a regression of Y versus a single X when an outlier is included versus when it is not included? Will this necessarily happen when a point is an outlier? Answer by giving a couple of examples.
The file P12_11.xlsx contains annual U.S. federal debt from 1960 through 2011. Fit an exponential growth curve to these data. Write a short report to summarize your findings. If the U.S. federal debt continues to rise at the exponential rate you find, approximately what will its value be at the end
The file P12_24.xlsx contains the daily closing prices of Procter & Gamble stock from July2014 to September 2015. Use only the 2014 data to estimate the trend component of the random walk model. Next, use the estimated random walk model to forecast the behavior of the time series for the 2015
The file P12_10.xlsx contains annual revenues for a convenience store. If you want to forecast revenue for the next few years with the moving averages method, what span should you use? Will any span work well?
The file P12_29.xlsx contains monthly revenues for a company over a four-year period. Use the moving average method to forecast this company's revenues for the 12 months. What span seems to work best?
Suppose that a time series consisting of six years (2011−2016) of quarterly data exhibits obvious seasonality. In fact, assume that the seasonal indexes turn out to be 0.75, 1.45, 1.25, and 0.55.a. If the last four observations of the series (the four quarters of 2016) are 2502, 4872, 4269, and
The file P12_51.xlsx contains monthly data on the nonfarm hire rate in the United States since 2005.a. What evidence is there that seasonality is important in this series? Find seasonal indexes (by any method you like) and state briefly what they mean.b. Forecast the next 12 months by using a
The file P12_59.xlsx contains revenue (in millions of dollars) for Procter & Gamble. Create a time series graph of these data. Then superimpose a trend line with Excel's Trendline option. Which of the possible Trendline options seems to provide the best fit? Using this option, what are your
The file P12_61.xlsx contains annual data on carbon dioxide (CO2) levels since 1959, measured at the Mauna Loa Observatory in Hawaii. Fit linear, exponential, and quadratic (polynomial of order 2) trends to these data. In terms of MAD, which fit is best? Using the best fit, forecast CO2 levels for
"A truly random series will likely have a very small number of runs." Is this statement true or false? Explain your choice.
Distinguish between a correlation and an autocorrelation. How are these measures similar? How are they different?
Under what conditions would you prefer a simple exponential smoothing model to the moving averages method for forecasting a time series?
Is it more appropriate to use an additive or a multiplicative model to forecast seasonal data? Summarize the difference(s) between these two types of seasonal models.
Suppose that monthly data on some time series variable exhibits a clear upward trend but no seasonality. You decide to use moving averages, with any appropriate span. Will there tend to be a systematic bias in your forecasts? Explain why or why not.
Suppose that monthly data on some time series variable exhibits obvious seasonality. Can you use moving averages, with any appropriate span, to track the seasonality well? Explain why or why not.
Suppose that quarterly data on some time series variable exhibits obvious seasonality, although the seasonal pattern varies somewhat from year to year. Which method do you believe will work best: Winters' method or regression with dummy variables for quarters (and possibly a time variable for
Most companies that use (any version of) exponential smoothing use fairly small smoothing constants such as 0.1 or 0.2. Explain why they don't tend to use larger values.
Suppose you use Solver to find the optimal solution to a maximization model. Then you remember that you omitted an important constraint. After adding the constraint and running Solver again, is the optimal value of the objective guaranteed to decrease? Why or why not?
In any optimization model such as those in this chapter, we say that the model is unbounded (and Solver will indicate as such) if there is no limit to the value of the objective. For example, if the objectiveis profit, then for any dollar value, no matter how large, there is a feasible solution
Consider an optimization model with a number of resource constraints. Each indicates that the amount of the resource used cannot exceed the amount available. Why is the shadow price of such a resource constraint always zero when the amount used in the optimal solution is less than the amount
If you add a constraint to an optimization model, and the previously optimal solution satisfies the new constraint, will this solution still be optimal with the new constraint added? Why or why not?
Why is it generally necessary to add nonnegativity constraints to an optimization model? Wouldn't Solver automatically choose nonnegative values for the decision variable cells?
Suppose you have a linear optimization model where you are trying to decide which products to produce to maximize profit. What does the additive assumption imply about the profit objective? What does the proportionality assumption imply about the profit objective? Be as specific as possible. Can
In a typical product mix model, where a company must decide how much of each product to produce to maximize profit, discuss possible situations where there might not be any feasible solutions. Could these be realistic? If you had such a situation in your company, how might you proceed?
In a typical product mix model, where a company must decide how much of each product to produce to maximize profit, there are sometimes customer demands for the products. We used upper-bound constraints for these: Don't produce more than you can sell. Would it be realistic to have lower-bound
In a production scheduling problem like Pigskin's, suppose the company must produce several products to meet customer demands. Would it suffice to solve a separate model for each product, as we did for Pigskin, or would one big model for all products be necessary? If the latter, discuss what this

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