## GET Practical Business Statistics TEXTBOOK SOLUTIONS

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Find the one-sided 95% confidence interval for the weight of candy bars after intervention, based on the data in Table 5.5.4, indicating that the population mean weight is no more than some amount.

From a list of the 729 people who went on a cruise, 130 were randomly selected for interview. Of these, 112 said that they were very happy with the accommodations. Find the 95% confidence interval for the population percentage who would have said they were very happy with the accommodations.

Suppose you learn that the p-value for a hypothesis test is equal to 0.0217. What can you say about the result of this test?

Suppose you have an estimator and would like to test whether or not the population mean value equals 0. What do you need in addition to the estimated value?

What standard error would you use to test whether a new observation came from the same population as a sample?

Part of the assembly line will need adjusting if the consistency of the injected plastic becomes either too viscous or not viscous enough as compared with a value (56.00) your engineers consider reasonable. You will decide to adjust only if you are convinced that the system is “not in control,” that is, there is a real need for adjustment. The average viscosity for 13 recent measurements was 51.22 with a standard error of 3.18.

a. Identify the null and research hypotheses for a two-sided test, using both words and mathematical symbols.

b. Perform a two-sided test at the 5% significance level and describe the result.

c. Perform a two-sided test at the 1% significance level and describe the result.

d. State the p-value as either p>0.05, p<0.05, p<0.01, or p<0.001.

e. Find the p-value using statistical software

Some of your advertisements seem to get no reaction, as though they are being ignored by the public. You have arranged for a study to measure the public’s awareness of your brand before and after viewing a TV show that includes the advertisement in question. You wish to see if the ad has a statistically significant effect as compared with zero, representing no effect. Your brand awareness, measured on a scale from 1 to 5, was found to have increased an average of 0.22 point when 200 people were shown an advertisement and questioned before and after. The standard deviation of the increase was 1.39 points.

a. Identify the null and research hypotheses for a two-sided test, using both words and mathematical symbols.

b. Perform a two-sided test at the 5% significance level and describe the result. c. Perform a two-sided test at the 1% significance level and describe the result.

d. State the p-value as either p>0.05, p<0.05, p<0.01, or p<0.001.

e. Find the p-value using statistical software.

Your factory’s inventory level was determined at 12 randomly selected times last year, with the following results: 313, 891, 153, 387, 584, 162, 742, 684, 277, 271, 285, 845

a. Find the typical inventory level throughout the whole year, using the standard statistical summary.

b. Identify the population.

c. Find the 95% confidence interval for the population means inventory level.

d. Is the average of the measured inventory levels significantly different from 500, which is the number used for management budgeting purposes? Justify your answer.

Your bakery produces loaves of bread with “1 pound” written on the label. Here are weights of randomly sampled loaves from today’s production: 1.02, 0.97, 0.98, 1.10, 1.00, 1.02, 0.98, 1.03, 1.03, 1.05, 1.02, 1.06

a. Find the 95% confidence interval for the mean weight of all loaves produced today.

b. Find the reference value for testing the average of the actual weights against the claim on the label.

c. Find the hypotheses, H0 and H1

d. Perform the hypothesis test (two-sided, level 0.05) and report the result. e. What error, if any, might you have committed?

e. What error, if any, might you have committed?

At a recent meeting, it was decided to go ahead with the introduction of a new product if “interested consumers would be willing, on average, to pay \$20.00 for the product.” A study was conducted, with 315 random interested consumers indicating that they would pay an average of \$18.14 for the product. The standard deviation was \$2.98.

a. Identify the reference value for testing the mean for all interested consumers.

b. Identify the null and research hypotheses for a two-sided test using both words and mathematical symbols.

c. Perform a two-sided test at the 5% significance level and describe the result.

d. Perform a two-sided test at the 1% significance level and describe the result.

e. State the p-value as either p>0.05, p<0.05, p<0.01, or p<0.001.

f. Find the p-value using statistical software.

The p-value is 0.0371. What conclusions can you reach and what error might have been made?

Do initial public offerings (IPOs) of stock significantly increase in value, on average, in the short term? Test using the data from Table 4.3.7 that show the performance of initial offerings as percent increases from the offer price, with most newly traded companies increasing in value while some lost money. Please give the p-value (as either p>0.05, p<0.05, p<0.01, or p<0.001) as part of your answer.

Test whether or not the population percentage could reasonably be 20%, based on the observed 18.4% who like your products, from a random sample of 500 consumers.

As part of a decision regarding a new product launch, you want to test whether or not a large enough percentage (10% or more) of the community would be interested in purchasing it. You will launch the product only if you find convincing evidence of such demand. A survey of 400 randomly selected people in the community finds that 13.0% are willing to try your proposed new product.

a. Why is a one-sided test appropriate here?

b. Identify the null and research hypotheses for a one-sided test using both words and mathematical symbols.

c. Perform the test at the 5% significance level and describe the result.

d. Perform the test at the 1% significance level and describe the result.

e. State the p-value as either p>0.05, p<0.05, p<0.01, or p<0.001.

You are considering a new delivery system and wish to test whether delivery times are significantly different, on average than your current system. It is well established that the mean delivery time of the current system is 2.38 days. A test of the new system shows that, with 48 observations, the average delivery time is 1.91 days with a standard deviation of 0.43 day.

a. Identify the null and research hypotheses for a two-sided test, using both words and mathematical symbols.

b. Perform a two-sided test at the 5% significance level and describe the result.

c. Perform a two-sided test at the 1% significance level and describe the result.

d. State the p-value as either p>0.05, p<0.05, p<0.01, or p<0.001.

e. Summarize the results in a brief memo to management

Here are the satisfaction scores given by 12 randomly selected customers: 89, 98, 96, 65, 99, 81, 76, 51, 82, 90, 96, 76 Does the observed average score differ significantly from the target score of 80? Justify your answer.

Production yields vary and can be high or low on a given day. If they are high, you want to find out why so that yields could be similarly increased on other days. If they are low, you want to fix the problem. You have just learned that today’s production yields seem to be lower than usual. Should you use a one-sided test or a two-sided test to investigate? Why?

A recent poll of 1,423 randomly sampled likely voters shows your favorite candidate ahead, with 51.93% in favor. There are two candidates. Use hypothesis testing to infer to the larger group of all likely voters to see whether or not this indicates that your candidate is ahead in the larger population.

a. Carefully identify the two-sided hypotheses.

b. Perform the hypothesis test at level 0.05 and give the result.

c. Make a careful, exact statement summarizing the result of the test and what it means.

d. Repeat parts b and c assuming that the percentage is 56.64% instead of 51.93%.

e. Explain why a one-sided test would be inappropriate here by showing that each of the three possible outcomes of a two-sided test would be of interest.

Managers perceived employee stock ownership as having a significant positive effect on product quality. As part of that same study, managers were also asked to rate the effect of employee stock ownership on unit labor cost.24 This effect, on a scale from 2 (large negative effect) to 2 (large positive effect), was 0.12 with a standard error of 0.11, based on a sample of 343 managers.

a. Find the 95% confidence interval and state carefully what this represents. Keep in mind that these are opinions of randomly selected managers.

b. Is there a significant relationship between employee stock ownership and the unit cost of labor as perceived by managers? Why or why not?

c. Identify the null and research hypotheses.

d. Which hypothesis has been accepted? Is this a weak or a strong conclusion?

e. Has the accepted hypothesis been absolutely proven? If not, what type of error may have been made?

You are supervising an audit to decide whether or not any errors in the recording of account transactions are “material errors.” Each account has a reported balance, whose accuracy can be verified only by careful and costly investigation; the account’s error is defined as the difference between the reported balance and the actual balance. Note that the error is zero for any account that is correctly reported. In practical terms, for this situation involving 12,000 accounts, the total error is material only if it is at least \$5,000. The average error amount for 250 randomly selected accounts was found to be \$0.25, and the standard deviation of the error amount was \$193.05. You may assume that your reputation as an auditor is on the line, so you want to be fairly certain before declaring that the total error is not material.

a. Find the estimated total error based on your sample and compare it to the material amount.

b. Identify the null and research hypotheses for a one-sided test of the population mean error per account and explain why a one-sided test is appropriate here.

c. Find the appropriate one-sided 95% confidence interval statement for the population mean error per account. d. Find the t-statistic.

e. Which hypothesis is accepted as a result of a onesided test at the 5% level?

f. Write a brief paragraph explaining the results of this audit.

Dishwasher detergent is packaged in containers that claim a weight of 24 ounces. Although there is some variation from one package to another, your policy is to ensure that the mean weight for each day’s production is slightly over 24 ounces. A random sample of 100 packages from today’s production indicates an average of 24.23 ounces with a standard deviation of 0.15 ounce.

a. Find the p-value (as either p>0.05, p<0.05, p<0.01, or p<0.001) for a one-sided hypothesis test to check if the population mean weight is above the claimed weight.

b. Write a brief paragraph summarizing your test and its results.

c. Is your conclusion a strong one or a weak one? Why?

Selected mutual funds that practice socially aware investing, with year-to-date rates of return, are shown in Table 10.7.3. On average, these funds lost value in the first half of 2010, in the sense that their average rate of return was negative. However, the Standard & Poor’s 500 stock market index lost 9.03% of its value during the same period, so this was a difficult time for the market in general.

a. On average, as a group, did socially aware mutual funds lose significantly more than the market index? Please use the market index as the reference value.

b. Find the p-value for this test (as either p>0.05, p<0.05, p<0.01, or p<0.001). In particular, is it highly significant?

c. Identify the underlying hypotheses and assumptions involved in part a.

d. Under these assumptions, the hypothesis test makes a clear and correct statement. However, are the assumptions realistic? Be sure to address independence (note that some of these funds are part of the same group).

e. Why is a two-sided test appropriate in this case?

World investments markets were highly volatile in 1998. Table 10.7.4 shows one-year rates of return on closedend mutual funds that specialize in income from international sources.

a. Do the rates of return of these closed-end world income funds, as a group, differ significantly on average from the 2.59% overall performance representing all world mutual funds over the same time period? If so, were these closed-end funds significantly better or significantly worse? In your calculations, you may assume that the overall performance is measured without randomness.

b. Do the rates of return of these closed-end world income funds, as a group, differ significantly on average from the 26.83% overall performance representing all emerging markets’ mutual funds over the same time period? If so, were these closed-end funds significantly better or significantly worse? In your calculations, you may assume that the overall performance is measured without randomness.

Your broker achieved a rate of return of 18.3% on your portfolio last year. For a sample of 25 other brokers in the area, according to a recent news article, the average rate of return was 15.2% with a standard deviation of 3.2% (as percentage points).

a. To test whether your broker significantly outperformed this group, identify the idealized population and the hypotheses being tested. In particular, are you testing against a mean or against a new observation?

b. Find the standard error for prediction.

c. Find the two-sided 95% prediction interval for a new observation.

d. Did your broker outperform this group?

e. Did your broker significantly outperform this group?

f. Find the t-value and the p-value (as either p>0.05, p<0.05, p<0.01, or p<0.001) for this two-sided test.

To understand your competitive position, you have examined the reliability of your product as well as the reliability of your closest competitor’s product. You have subjected each product to abuse that represents about a year’s worth of wear-and-tear per day. Table 10.7.7 shows the data indicating how long each item lasted.

a. Find the average time to failure for your and your competitor’s products. Find the average difference (yours minus your competitor’s).

b. Find the appropriate standard error for this average difference. In particular, is this a paired or an unpaired situation? Why?

c. Find the two-sided 99% confidence interval for the mean difference in reliability.

d. Test at the 1% level if there is a significant difference in reliability between your products and your competitor’s at this test level.

e. Find the p-value for the difference in reliability (as either p>0.05, p<0.05, p<0.01, or p<0.001) and state whether or not the result is significant at the conventional test level.

f. Write a brief paragraph, complete with footnote(s) that might be used in an advertising brochure showing off your products.

You are analyzing the results of a consumer survey of a product, rated on a scale from 1 to 10. For the 130 consumers who described themselves as “outgoing,” the average rating was 8.36, and the standard deviation was 1.82. For the 218 “shy” consumers, the average was 8.78, and the standard deviation was 0.91.

a. Test to see if there is a significant difference between the ratings of outgoing and shy consumers.

b. Report the test results using p-value notation (as either p>0.05, p<0.05, p<0.01, or p<0.001).

A cup of coffee is found to have only 72.8 milligrams of caffeine. Test (at the 5% level) whether the beans used could have come from the same population as those that generated the data in problem 47 of Chapter 9.

Problem 47

Consider the weights for two samples of candy bars, before and after intervention, from Table 5.5.4.

Distinguish correlation and regression analysis.

Distinguish correlation and regression analysis.

a. Which is usually better, a lower or a higher value for R2

b. Which is better, a lower or a higher value for Se?

What is extrapolation? Why is it especially troublesome?

A linear regression analysis has produced the following equation relating profits to hours of managerial time spent developing the past year’s projects at a firm: Profits = -\$957 + \$85Number of hours

a. According to this estimated relationship, how large would the profits (or losses) be if no time were spent in planning?

b. On the average, an extra 10 h spent planning resulted in how large an increase in project profits?

c. Find the break-even point, which is the number of hours for which the estimated profits would be zero.

d. If the correlation is r = 0.351, what percentage of the variation in profits is explained by the time spent?

e. How much of the variation in profits is left unexplained by the number of hours spent? Write a paragraph explaining how much faith you should have in this prediction equation and discussing other factors that might have an impact on profits.

Consider the number of transactions and the total dollar value of merger and acquisition deals in the oil and gas industry, from Table 11.1.6.

a. Find the regression equation for predicting the dollar value from the number of transactions.

b. What is the estimated dollar value attributable to a single additional transaction for these investment bankers, on average?

c. Draw a scatterplot of the data set with the regression line.

d. Find the expected dollar amount for Goldman Sachs and the residual value. Interpret both of these values in business terms.

e. Find the standard error of the slope coefficient. What does this number indicate?

f. Find the 95% confidence interval for the expected marginal value of an additional transaction to these firms. (This is economics language for the slope.)

g. Test at the 5% level to see if there is a significant relationship between the number of transactions and the dollar value.

h. Your investment banking firm is aiming to be in the top group next year, with 25 transactions. Assuming that you will be “just like the big ones,” compute a 95% confidence interval for the dollar amount you will handle.

In the territory versus sales example (based on the data from Table 11.2.3), the least-squares line to predict sales based on the population of the territory was found to be Expected sales = \$1,371,744 + \$0:23675045 (Population)

a. Interpret the slope coefficient as a number with a simple and direct business meaning.

b. What proportion of the variation in sales from one agent to another is attributable to territory size? What proportion is due to other factors?

c. Does territory size have a significant impact on sales? How do you know?

d. Find the p-value (as either p>0.05, p<0.05, p<0.01, or p<0.001) for the significance of the slope coefficient.

e. Find the actual p-value for the significance of the slope coefficient, using statistical software, and use it to verify your answer to part d.

The least-squares prediction equation is, predicted costs = 35.2 + 5.3 (items), with predicted costs measured in dollars. Find the predicted value and residual for a situation with costs of \$600 and 100 items.

For each of the scatterplots in Figs. 11.3.1–11.3.4, say whether the correlation is closest to 0.9, 0.5, 0.0, -0.5, or -0.9.

Given the correlation r = -0.603 and the least-squares prediction equation Y = 38.2–5.3X, find the predicted value for Y when X is 15.

Given the correlation r = 0.307 and the least-squares prediction equation Y = 55.6 + 18.2X, find the predicted value for Y when X is \$25.

One day your factory used \$385 worth of electricity to produce 132 items. On a second day, \$506 worth of electricity was consumed to produce 183 items. On a third day, the numbers were \$261 and 105. How much electricity do you estimate it would take to produce 150items?

Which of the following correlation coefficients corresponds to a moderately strong relationship with higher X values associated with higher Y values: r = 1, r = 0.73, r = 0.04, r = –0.83, or r = -0.99?

On Monday, your business produced 7 items which cost you \$18. On Tuesday, you produced 8 costing \$17. On Wednesday, you produced 18 costing \$32. On Thursday, you produced 3 items costing \$16. Using a linear regression model accounting for fixed and variable costs, give your estimate of Friday’s costs for producing 10 items.

Identify the structure of the scatterplot in Fig. 11.3.5.

Identify the structure of the scatterplot in Fig. 11.3.6.

Why should variables measured in the same basic units be transformed in the same way?

Your firm is wondering about the results of magazine advertising as part of an assessment of the marketing strategy. For each ad, you have information on its cost, its size, and the number of inquiries it generated. In particular, you are wondering if the number of leads generated by an ad has any connection with its cost and size. Identify the Y variable, the X variables, and the appropriate statistic or test.

It is budgeting time again, and you would like to know the expected payoff (in terms of dollars collected) of spending an extra dollar on collection of delinquent accounts, after adjusting for the size of the pool of delinquent accounts. Identify the Y variable, the X variables, and the appropriate statistic or test.

In order to substantiate a claim of damages, you need to estimate the revenues your firm lost when the opening of the new lumber mill was delayed for three months. You have access to data for similar firms on their total assets, lumber mill capacity, and revenues. For your firm, you know total assets and lumber mill capacity (if it had been working), but you wish to estimate the revenues. Identify the Y variable, the X variables, and the appropriate statistic or test.

Productivity is a concern. For each employee, you have data on productivity as well as other factors. You want to know how much these factors explain about the variation in productivity from one employee to another. Identify the Y variable, the X variables, and the appropriate statistic or test.

In a multiple regression, what would you suspect is the problem if the R2 is large and significant, but none of the X variables has a t test that is significant?

What is the primary purpose of writing a report?

How can you use the executive summary and introduction to reach a diverse audience with limited time?

Should you assume that everyone who reads your conclusion is already familiar with all of the details of the analysis and methods section?

What can you do to help those in your audience who are short of time?

When is the best time to write the introduction and executive summary, first or last? Why?

What is the relationship between the outline and the finished report?

How would you check the meaning of a word to be sure that you are using it correctly?

How can you find synonyms for a given word? Why might you want to?

What important information is missing from each of the following references?

a. Personal communication, 2016.

c. Basic Business Communication (Burr Ridge, Ill.: Richard D. Irwin).

d. James A. White, “Will the Real S&P 500 Please Stand Up? Investment Firms Disagree on Index,” Wall Street Journal.

e. Data were obtained from the White House Economic Statistics Briefing Room on the Internet.

Distinguish stationary and nonstationary time-series behavior.

What kinds of additional terms are needed to include seasonal behavior in advanced ARIMA models?

Which time-series method of analysis would be most appropriate to a situation in which forecasts and confidence limits are needed for a data set that shows medium-term cyclic behavior?

Which time-series method of analysis would be most appropriate to a situation in which prices are lower at harvest time in the fall but are typically higher the rest of the year and in which there is a need for a methodology that is relatively easy to understand?

Consider the Walt Disney Company’s quarterly revenues as shown in Table 14.4.1.

a. Draw a time-series plot for this data set. Describe any trend and seasonal behavior that you see.

b. Find the moving average values and plot them on the same graph as the original data. Comment on what you see.

c. Find the seasonal index for each quarter. In particular, how much higher is the fourth quarter than a typical quarter during the year?

d. Find the seasonally adjusted values and plot them with the original data. Comment on what you see.

e. From fourth quarter 2014 to the first quarter of 2015, revenues fell from 13.391 to 12.461. What happened on a seasonally adjusted basis?

f. Find the regression equation to predict the long-term trend in seasonally adjusted sales for each time period, using 1, 2,… for the X variable.

g. Compute the seasonally adjusted forecast for the fourth quarter of 2017.

h. Compute the forecast for the first quarter of 2018.

Consider Intel’s Net Revenue in Table 14.4.2.

a. Construct a time-series plot for this data set. Describe the seasonal and cyclic behavior that you see, as well as any evidence of irregular behavior.

b. Which quarter (1, 2, 3, or 4) appears to be Intel’s best in terms of net revenue, based on your plot in part a?

c. Is the seasonal pattern (in your graph for part a) consistent across the entire time period?

d. Calculate the moving average (using 1 year of data at a time) for this time series. Construct a time-series plot with both the data and the moving average.

e. Describe the cyclic behavior revealed by the moving average.

f. Find the seasonal index for each quarter. Do these values appear reasonable compared to the time-series plot of the data?

g. Find the seasonally adjusted sales corresponding to each of the original sales values. Construct a time-series plot of this seasonally adjusted series.

h. Do you see an overall linear long-term trend up or down throughout these sales data? Would it be appropriate to use a regression line for forecasting this series?

i. Intel’s revenue rose from 14.554 to 14.721 billion from the third to the fourth quarter of 2014. What happened to revenue on a seasonally adjusted basis?

Table 14.4.3 shows the quarterly net sales of Mattel, a major designer, manufacturer, and marketer of toys. Because of seasonal gift giving, you might expect fourth-quarter sales to be much higher, generally, than those of the other three quarters of the year.

a. Construct a time-series plot for this data set. Describe any trend and seasonal behavior that you see in the plot.

b. Calculate the moving average (using 1 year of data at a time) for this time series. Construct a time-series plot with both the data and the moving average.

c. Find the seasonal index for each quarter. Do these values appear reasonable when you look at the time-series plot of the data?

d. Which is Mattel’s best quarter (1, 2, 3, or 4)? On average, how much higher are sales as compared to a typical quarter during the year?

e. Find the seasonally adjusted sales corresponding to each of the original sales values.

f. From the second to the third quarter of 2014, sales went up from 1.062 to 2.021 billion. What happened on a seasonally adjusted basis?

g. From the first to the second quarter of 2014, Mattel’s sales rose by over \$100 million, from 0.946 to 1.062 billion. What happened on a seasonally adjusted basis?

h. Find the regression equation to predict the long-term trend in seasonally adjusted sales for each time period, using 1, 2,… for the X variable.

i. Compute the seasonally adjusted forecast for the second quarter of 2015.

j. Compute the forecast for the second quarter of 2015.

k. Compare the forecast from part j to Mattel’s actual net sales of 0.988 billion for the second quarter of 2015. Is your result consistent with the possibility that the strengthening dollar during this time period reduced the value of foreign sales?

Amazon.com is an e-commerce firm that has shown considerable growth since its founding in 1995, and its quarterly net sales are shown in Table 14.4.4. Their 2014 annual report includes a section titled “Seasonality” that states: “Our business is affected by seasonality, which historically has resulted in higher sales volume during our fourth quarter, which ends Dec. 31. We recognized 33%, 34%, and 35% of our annual revenue during the fourth quarter of 2014, 2013, and 2012.”

a. Construct a time-series plot for this data set. Do you agree that there are seasonal factors present here?

b. Calculate the moving average (using 1 year of data at a time) for this time series. Construct a time-series plot with both the data and the moving average.

c. Describe any cyclic behavior that you see in the moving average.

d. Find the seasonal index for each quarter. Do these values appear reasonable when you look at the time-series plot of the data?

e. Which is Amazon.com’s best quarter (1, 2, 3, or 4)? On average, how much higher are sales then as compared to a typical quarter during the year?

f. Which is Amazon.com’s worst quarter (1, 2, 3, or 4)? On average, how much lower are sales then as compared to a typical quarter during the year?

g. Find the seasonally adjusted sales corresponding to each of the original sales values. Construct a timeseries plot of this seasonally adjusted series.

h. Describe the behavior of the seasonally adjusted series. In particular, identify any variations in growth rate that are visible over this time period.

Consider PepsiCo’s quarterly net revenue as shown in Table 14.4.5.

a. Draw a time-series plot for this data set. Describe any trend and seasonal behavior that you see.

b. Plot the moving average values on the same graph as the original data. Comment on what you see.

c. Find the seasonal index for each quarter. Which is generally the best quarter for PepsiCo? About how much larger are net sales in this quarter, as compared to a typical quarter?

d. Plot the seasonally adjusted series with the original data. e. Find the regression equation to predict the long-term trend in seasonally adjusted sales for each time period, using 1, 2,… for the X variable.

f. Does PepsiCo show a significant trend (either up or down) over this time period as indicated by the regression analysis in the previous part of this problem?

g. If we omit the first year (the four observations in 2010) but still use the other seasonally adjusted values as we did in the previous regression, does PepsiCo show a significant trend (either up or down) over this time period?

The number of diners per quarter eating at your apres-ski restaurant has been examined using trend-seasonal analysis. The quarterly seasonal indexes are 1.45, 0.55, 0.72, and 1.26 for quarters 1, 2, 3, and 4, respectively. A linear trend has been estimated as 5,423 + 408 (Quarter number), where the quarter number starts at 1 in the first quarter of 2012 and increases by 1 each successive quarter.

a. Find the seasonally adjusted forecast value for the first quarter of 2019.

b. Find the seasonally adjusted forecast value for the second quarter of 2019.

c. Why is the seasonally adjusted forecast larger in the second quarter, in which you would expect fewer skiers coming to dinner?

d. Find the forecast value for the first quarter of 2019.

e. Find the forecast value for the second quarter of 2019.

f. On a seasonally adjusted basis, according to this estimated linear trend, how many more diners do you expect to serve each quarter compared to the previous quarter?

g. Your strategic business plan includes a major expansion project when the number of diners reaches 80,000 per year. In which calendar year will this first happen, according to your forecasts?

Consider the time series of quarterly sales in thousands shown in Table 14.4.6. The seasonal indices are 0.89 for quarter 1, 0.88 for 2, 1.27 for 3, and 0.93 for 4.

a. Find the seasonally adjusted sales corresponding to each sales value.

b. In which quarter is the most business generally done?

c. As indicated in the data, sales increased from 817 to 1,073 in 2015 from quarters 2 to 3. What happened during this period on a seasonally adjusted basis?

d. As indicated in the data, sales decreased from 1,084 to 819 in 2014 from quarters 3 to 4. What happened during this period on a seasonally adjusted basis?

e. The exponential trend values for the four quarters of 2019 are 1,964, 2,070, 2,183, and 2,301. Seasonalize these trend forecasts to obtain actual sales forecasts for 2019.

Which type of time-series analysis would provide the simplest results for studying the demand for heating oil, which tends to be highest in the winter?

Your seasonally adjusted monthly sales forecast is \$382,190 +\$4,011 (Month number), where the month number is 1 for Jan. 2011 and increases by 1 each month. The seasonal index for February sales is 0.923, and it is 1.137 for April. What you need now is a forecast for cost of goods sold in order to plan ahead for filling future orders. You have found that monthly sales have been a good predictor of monthly cost of goods sold and have estimated the following regression equation: Predicted cost of good sold = \$106,582 + 0:413 (Sales)

a. Find the seasonally adjusted forecast of monthly sales for Feb. 2018.

b. Find the forecast of monthly sales for Feb. 2018.

c. Find the forecast of cost of goods sold for Feb. 2018.

d. Find the forecast of cost of goods sold for Apr. 2019.

Gross Domestic Product (GDP) is an important measure of total production and is used by business to help guide their planning for the future. Table 14.4.8 shows a BoxJenkins analysis of the percentage change in GDP (from the same quarter of the previous year, as a measure of the growth rate of the overall economy), while Fig. 14.4.1 shows the data series with the Box-Jenkins forecasts.20

a. What kind of process has been estimated?

b. Which estimated coefficients (if any) are significant?

c. Based on the figure, would you be surprised if GDP fell by 5 percentage points (as compared to the same quarter in the previous year) in the year 2018?

d. Based on the figure, would you be surprised if GDP grew by 4 percentage points (as compared to the same quarter in the previous year) in the year 2019?

e. The forecasts in the figure appear to level off after about 2018. Does this tell you that the GDP growth rate will stop changing from year to year in the future? Explain.

Fig. 14.4.1

The number of job openings fluctuates through time, providing useful information about the current state of the economy and possibilities for the future. Table 14.4.11 shows the computed results of a BoxJenkins analysis of job openings in thousands, annually at the start of each year from 2001 to 2015, while Fig. 14.4.2 shows the data series with the Box-Jenkins forecasts.21

a. What kind of component (autoregressive or moving average) does the estimated model include?

b. How many differences are used in the model?

c. Is the model component that you identified in part a significant?

d. Is the constant term significant?

e. Based on the figure, would you be surprised to see 15,000,000 job openings in 2020?
f. Based on the figure, would you be surprised to see 7,500,000 job openings in 2020?

Which assumption helps the data be representative of the population?

What assumptions are required concerning the distribution of each population?

Do the sample sizes have to be equal in the one-way analysis of variance?

Camera angle can make a difference in advertising; it can even affect the viewer’s evaluation of a product. A research article reported a main effect for camera angle (F2,29 = 14.48, p<0.001) based on an analysis of variance. The average score was 4.51 for eye-level camera angle, 5.49 for a low-angle looking up, and 3.61 for a high angle looking down. Higher scores represent more positive evaluations of the product (a personal computer). Are there significant differences among these three camera angles? If so, which angle appears to be best?

Another experiment in the report by Meyers-Levy and Peracchio involved the evaluation of bicycle pictures taken with various camera angles, as evaluated by two groups of individuals with different levels of motivation. (The high-motivation group believed they had a reasonable chance to win a bicycle.) Evaluation scores, on average, were higher when the camera angle was upward or at eye level, and lower when the bicycle was viewed looking down. These differences were larger for the low-motivation group. The ANOVA results of the evaluation scores included an examination of the main effect for camera angle (F2,106 = 7.00, p<0.001), the main effect for motivation (F1,106 = 3.78, p<0.05), and their interaction (F2,106 = 3.83, p<0.03).

a. Are there significant differences in the average evaluation scores of the low-motivation and the high motivation groups? Justify your answer.

b. Does the information provided here from the analysis of variance tell you whether it was the low motivation group or the high-motivation group that gave higher evaluations, on average?

c. Is there a significant interaction between camera angle and motivation? Justify your answer.

d. Can you conclude that the camera angle makes more of a difference when marketing to the low-motivation group than to the high-motivation group, or are the effects of camera angle basically similar for the two groups, except for randomness? Explain your answer.

Would there be any problem with a nonparametric analysis (two unpaired samples) of data in Table 10.7.8 listing day care rates comparing those of the well-to-do Laurelhurst area to other parts of Seattle? Why or why not?

Are tasting scores significantly different for the Chardonnay and Cabernet-Sauvignon wines listed in Table 10.7.6? Is this a paired or unpaired situation?

The number of items returned for each of the past 9 days was 13, 8, 36, 18, 6, 4, 39, 47, and 21. Test to see if the median number returned is significantly different from 40 and find the p-value (as either p>0.05, p<0.05, or p<0.01).

Perform a nonparametric analysis of prescription drug prices in the United States and Canada, as reported in Table 16.4.7.

a. Is this a paired or unpaired situation?

b. Are prices significantly higher in the United States? How do you know?

What is the purpose of the chi-squared test for equality of percentages?

What is the purpose of the chi-squared test for independence?

Are your customers special? In particular, is their interest level in your promotional newsletter higher than for potential customers (who are not currently customers)? Justify your answer by reporting the chi-squared statistic and its degrees of freedom for the data set reported in Table 17.4.8 based on a random sample for each group.

a. What is the purpose of the X̅ chart?

b. What is a typical sample size?

c. How would you find the center line if you had no standard?

d. How would you find the center line if you did have a standard?

e. How would you find the control limits if you had no standard?

f. How would you find the control limits if you did have a standard?

What problems, if any, are visible in the control charts in Fig. 18.5.3? What action (if any) would you suggest?

Fig 18.5.3

What problems, if any, are visible in the control charts in Fig. 18.5.4? What action (if any) would you suggest?

Fig 18.5.4

What problems, if any, are visible in the control charts in Fig. 18.5.5? What action (if any) would you suggest?

Fig 18.5.5

For the genders:

a. Summarize by finding the percent of each category.

b. Find the mode. What does this tell you?

Appendix A

For the previous problem, compare the coefficient of variation before and after the price increase. Why does it change (or not change) in this way?.

Previous Problem

Your costs had been forecast as having an average of \$138,000 with a standard deviation of \$35,000. You have just learned that your suppliers are raising prices by 4% across the board.

For the preceding problem, compare the coefficient of variation before and after the sales goal adjustment. Why does it change (or not change) in this way?

Preceding problem

You are the sales manager for a regional division of a beverage company. The sales goals for your representatives have an average of \$768,000 with a standard deviation of \$240,000. You have been instructed to raise the sales goal of each representative by \$85,000.

Suppose two events are independent. One event has probability 0.27, while the other has probability 0.64.

a. Find the probability that both events happen.

b. Find the probability of the union of these events.

c. Find the probability that neither event happens.

Viewing the database in Appendix A as a random sample from a much larger population, consider the percentage of women. Find the 95% confidence interval.

Appendix A.

a. Perform a two-sided test at the 1% significance level for the previous problem and describe the result.

b. State the p-value as either p>0.05, p<0.05, p<0.01, or p<0.001.

c. Find the p-value using statistical software.

For the previous problem, interpret the regression coefficient for labor by estimating the average final cost associated with each dollar that management identified ahead of time as being labor related.

Previous Problem.

Consider the computer output in Table 12.5.12, part of an analysis to explain the final cost of a project based on management’s best guess of labor and materials costs at the time the bid was placed, computed from 25 recent contracts. All variables are measured in dollars.

Is it OK to repeat material in the introduction that already appeared in the executive summary?

Refer to the data for problem 5.

a. What does the null hypothesis of independence claim, in practical terms, for this situation?

b. How many managers responding “Worse” would you expect to find in this sample if response were independent of employee classification?

c. Find the table of expected counts, assuming independence.

d. Find the chi-squared statistic.

e. How many degrees of freedom does the chi-squared test have?

Data from Problem 5

Your firm is considering expansion to a nearby city. A survey of employees in that city, asked to respond to the question “Will business conditions in this area get better, stay the same, or get worse?” produced the data set shown in Table 17.4.3.

Refer to the data for problem 8.

a. Find the critical value from the chi-squared table for the 5% level and report the result of the chi-squared test.

b. Find the critical value from the chi-squared table for the 1% level and report the result of the chi-squared test.

c. Find the critical value from the chi-squared table for the 0.1% level and report the result of the chi-squared test.

d. State your conclusions (with p-value reported as p>0.05, p<0.05, p<0.01, or p<0.001) and discuss the results in practical terms.

Data from Problem 8

Consider the results of a small opinion poll concerning the chances of another stock market crash in the next 12 months comparable to the crash of 1987, shown in Table 17.4.4.

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