Business Forecasting Study Set 1

Business

Quiz 5 :
Simple Linear Regression

Quiz 5 :
Simple Linear Regression

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Which of the following situations is inconsistent? a. ? = 499 +.21 X and r =.75 b. ? = 100 +. 9 X and r = ?.70 c. ? = ?20 + 1 X and r =.40 d. ? = ?7 ? 4X and r = ?.90
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Answer:

Identify which of the given situations is inconsistent.
Incorrect option: Option (a)
img and
img The above situation is consistent.
Reason:
Here, the regression coefficient is
img and the correlation coefficient is
img which has same sign.
Incorrect option: Option (c)
img and
img The above situation is consistent.
Reason:
Here, the regression coefficient is
img and the correlation coefficient is
img which have same sign.
Incorrect option: Option (d)
img and
img The above situation is consistent.
Reason:
Here, the regression coefficient is
img and the correlation coefficient is
img which have same sign.
Correct option: Option (b):
img and
img The above situation is inconsistent.
Reason:
From the above equation, the regression coefficient is
img and the correlation coefficient is
img which have different sign.

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AT T (American Telegraph and Telephone) earnings in billions of dollars are estimated using GNP (gross national product). The regression equation is ? =.078 +.06 X , where GNP is measured in billions of dollars. a. Interpret the slope. b. Interpret the Y -intercept.
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Answer:

a.
Interpret the slope
img of the regression equation.
The regression equation is,
img Here,
img = Earnings, in billions of dollars and
img = GNP (gross national product), in billions of dollars.
From the regression equation, the slope
img is
img .
Interpretation:
The estimated earnings will increase by 0.06 billion of dollars as the GNP is increased by 1 billion of dollars. In other words, for each increase of 1 billion of dollars in the GNP increases the estimated earnings by 0.06 billion of dollars.
b.
Interpret the Y -intercept
img of the regression equation.
From the above regression equation, the Y -intercept
img is
img .
Interpretation:
The estimated earnings will be 0.078 billion of dollars when the GNP is 0 billion of dollars.

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Consider the data in Table P-3 where X = weekly advertising expenditures and Y = weekly sales. a. Does a significant relationship exist between advertising expenditures and sales? b. State the prediction equation. c. Forecast sales for an advertising expenditure of $50. d. What percentage of the variation in sales can be explained with the prediction equation? e. State the amount of unexplained variation. f. State the amount of total variation. TABLE P-3 img
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a.
Check whether or not there is a significant relationship between weekly advertising expenditures ( X ) and weekly sales ( Y ).
The hypotheses are given below:
Null hypothesis:
img That is, there is no linear relationship between weekly advertising expenditures and weekly sales.
Alternative hypothesis:
img That is, there is a linear relationship between weekly advertising expenditures and weekly sales.
Obtain the prediction equation by using the MINITAB.\
MINITAB procedure:
Step 1: Choose Stat Regression Regression.
Step 2: In the Response box , enter the Y variable as Sales from the box on the left.
Step 3: In the Predictors box , enter the X variable as Expend from the box on the left.
Step 4: Click OK.
MINITAB output:
img From the above MINITAB output, the value of F -ratio is
img and the corresponding p -value is
img .
Conclusion:
Here,
img is 0.002, which is lesser than any level of significance
img .
Therefore, by the rejection rule, reject the null hypothesis
img .
Thus, it can be concluded that there is an evidence to reject the null hypothesis
img at any level of significance
img .
Hence, there is a linear relationship between weekly advertising expenditures and weekly sales.
b.
State the prediction equation.
From the above MINITAB output, the prediction equation is
img …… (1)
Here,
img = Sales,
img = Expenditure
c.
Forecast the sales for an advertising expenditure of $50.
Substitute
img in equation (1)
img Thus, the forecasted sales for an advertising expenditure of $50 is
img .
d.
Obtain the percentage of the variation in sales can be explained with the prediction equation.
From the above MINITAB output, the coefficient of determination,
img is 0.719. Thus,
img of variation in sales can be explained with the prediction equation.
e.
Obtain the amount of unexplained variation.
From the above MINITAB output, the value of SSE is 36,121. Thus, the amount of unexplained variation is
img .
f.
Obtain the amount of total variation.
From the above MINITAB output, the value of SST is 128,552. Thus, the amount of total variation is
img .

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The times required to check out customers in a supermarket and the corresponding values of purchases are shown in Table P-4. Answer parts a, b, e, and f of Problem 3 using these data. Give a point forecast and a 99% interval forecast for Y if X = 3.0. TABLE P-4 img
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Lori Franz, maintenance supervisor for the Baltimore Transit Authority, would like to determine whether there is a positive relationship between the annual maintenance cost of a bus and its age. If a relationship exists, Lori feels that she can do a better job of forecasting the annual bus maintenance budget. She collects the data shown in Table P-5. a. Plot a scatter diagram. b. What kind of relationship exists between these two variables? c. Compute the correlation coefficient. d. Determine the least squares line. e. Test for the significance of the slope coefficient at the.05 significance level. Is the correlation significant? Explain. f. Forecast the annual maintenance cost for a five-year-old bus. TABLE P-5 img
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Andrew Vazsonyi is the manager of the Spendwise supermarket chain. He would like to be able to forecast paperback book sales (books per week) based on the amount of shelf display space (feet) provided. Andrew gathers data for a sample of 11 weeks, as shown in Table P-6. a. Plot a scatter diagram. b. What kind of relationship exists between these two variables? c. Compute the correlation coefficient. d. Determine the least squares line. e. Test for the significance of the slope coefficient at the.10 significance level. Is the correlation significant? Explain. f. Plot the residuals versus the fitted values. Based on this plot, is the simple linear regression model appropriate for these data? g. Forecast the paperback book sales for a week during which four feet of shelf space is provided. TABLE P-6 img
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Information supplied by a mail-order business for 12 cities is shown in Table P-7. a. Determine the fitted regression line. b. Calculate the standard error of the estimate. c. Determine the ANOVA table. d. What percentage of the variation in mail orders is explained by the number of catalogs distributed? e. Test to determine whether the regression slope coefficient is significantly different from zero. (Use the.01 significance level.) f. Test for the significance of the regression using the F statistic from the ANOVA table. (Use the.01 significance level.) Is the result consistent with the result in part e? Should it be? g. Forecast the number of mail orders received when 10,000 catalogs are distributed with a 90% prediction interval. TABLE P-7 img
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In a regression of investment on the interest rate, the results in Table P-8 were observed over a 10-year period. a. Is the relationship between these variables significant? b. Can an effective prediction equation be developed? c. If the average interest rate is 4% five years from now, can yearly investment be forecast? d. Calculate and interpret r 2. e. Discuss correlation and causation in this example. TABLE P-8 img
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The ABC Investment Company is in the business of making bids on investments offered by various firms that want additional financing. ABC has tabulated its bids on the last 25 issues in terms of their percentage of par value. The bids of ABC's major competitor, as a percentage of par value, are also tabulated on these issues. ABC now wonders if it is using the same rationale as its competitor in preparing bids. In other words, could ABC's bid be used to forecast the competitor's bid? If not, then the competitor must be evaluating issues differently. The data are given in Table P-9. TABLE P-9 img a. To what extent are the two firms using the same rationale in preparing their bids? b. Forecast the competitor's bid if ABC bids 101% of par value. Give both a point forecast and an interval prediction. c. Under part b, what is the probability of ABC winning this particular bid (the lowest bid wins)?
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Evaluate the following statements: a. A high r 2 means a significant regression. b. A very large sample size in a regression problem will always produce useful results.
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Ed Bogdanski, owner of the American Precast Company, has hired you as a part-time analyst. Ed was extremely pleased when you determined that there is a positive relationship between the number of building permits issued and the amount of work available to his company. Now he wonders if it is possible to use knowledge of interest rates on first mortgages to predict the number of building permits that will be issued each month. You collect a random sample of nine months of data, as shown in Table P-11. TABLE P-11 img a. Plot the data as a scatter diagram. b. Determine the fitted regression function. c. Test for the significance of the slope coefficient at the.05 significance level. d. When the interest rate increases by 1%, what is the average decrease in the number of building permits issued? e. Compute the coefficient of determination. f. Write a sentence that Ed can understand, interpreting the number computed in part e. g. Write Ed a memo explaining the results of your analysis.
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Consider the population of 140 observations that are presented in Table P-12. The Marshall Printing Company wishes to estimate the relationship between the number of copies produced by an offset printing technique ( X ) and the associated direct labor cost ( Y ). Select a random sample of 20 observations. a. Construct a scatter diagram. b. Compute the sample correlation coefficient. c. Determine the fitted regression line. d. Plot the fitted line on the scatter diagram. e. Compute the standard error of estimate. f. Compute the coefficient of determination and interpret its value. g. Test the hypothesis that the slope, ? 1 , of the population regression line is zero. h. Calculate a point forecast and a 90% prediction interval for the direct labor cost if the project involves 250 copies. i. Examine the residuals. Does it appear as if a simple linear regression model is appropriate for these data? Explain. TABLE P-12 img
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Harry Daniels is a quality control engineer for the Specific Electric Corporation. Specific manufactures electric motors. One of the steps in the manufacturing process involves the use of an automatic milling machine to produce slots in the shafts of the motors. Each batch of shafts is tested, and all shafts that do not meet required dimensional tolerances are discarded. The milling machine must be readjusted at the beginning of each new batch because its cutter head wears slightly during production. Harry is assigned the job of forecasting how the size of a batch affects the number of defective shafts in the batch so that he can select the best batch size. He collects the data for the average batch size of 13 batches shown in Table P-13 and assigns you to analyze it. a. Plot the data as a scatter diagram. b. Fit a simple linear regression model. c. Test for the significance of the slope coefficient. d. Examine the residuals. e. Develop a curvilinear model by fitting a simple linear regression model to some transformation of the independent variable. f. Test for the significance of the slope coefficient of the transformed variable. g. Examine the residuals. h. Forecast the number of defectives for a batch size of 300. i. Which of the models in parts b and e do you prefer? j. Write Harry a memo summarizing your results. TABLE P-13 img
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The data in Table P-14 were collected as part of a study of real estate property evaluation. The numbers are observations on X = assessed value (in thousands of dollars) on the city assessor's books and Y = market value (selling price in thousands of dollars) for n = 30 parcels of land that sold in a particular calendar year in a certain geographical area. TABLE P-14 img a. Plot the market value against the assessed value as a scatter diagram. b. Assuming a simple linear regression model, determine the least squares line relating market value to assessed value. c. Determine r 2 and interpret its value. d. Is the regression significant? Explain. e. Predict the market value of a property with an assessed value of 90.5. Is there any danger in making this prediction? f. Examine the residuals. Can you identify any observations that have a large influence on the location of the least squares line?
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Player costs ( X ) and operating expenses ( Y ) for n = 26 major league baseball teams for the 1990-1991 season are given in Table P-15. TABLE P-15 img a. Assuming a simple linear regression model, determine the equation for the fitted straight line. b. Determine r 2 and comment on the strength of the linear relation. c. Test for the significance of the regression with a level of significance of.10. d. Can we conclude that, as a general rule, operating expenses are about twice player costs? Discuss. e. Forecast operating expenses, with a large-sample 95% prediction interval, if player costs are $30.5 million. f. Using the residuals as a guide, identify any unusual observations. That is, do some teams have unusually low or unusually high player costs as a component of operating expenses?
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Table P-16 contains data for 23 cities on newsprint consumption ( Y ) and the number of families in the city ( X ) during a particular year. a. Plot newsprint consumption against number of families as a scatter diagram. b. Is a simple linear regression model appropriate for the data in Table P-16? Be sure your answer includes an analysis of the residuals. c. Consider a log transformation of newsprint consumption and a simple linear regression model relating Y = log (newsprint consumption) to X = number of families. Fit this model. d. Examine the residuals from the regression in part c. Which model, the one in part b or the one in part c, is better? Justify your answer. e. Using the fitted function in part c, forecast the amount of newsprint consumed in a year if a city has 10,000 families. f. Can you think of other variables that are likely to influence the amount of newsprint consumed in a year? TABLE P-16 img
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Outback Steakhouse grew explosively during its first years of operation. The numbers of Outback Steakhouse locations for the period 1988-1993 are given below. img Source: Outback Steakhouse. a. Does there appear to be linear or exponential growth in the number of steakhouses? b. Estimate the annual growth rate for Outback Steakhouse over the 1988-1993 period. c. Forecast the number of Outback locations for 2007. Does this number seem reasonable? Explain.
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On The Double, a chain of campus copy centers, began operations with a single store in 1993. The number of copy centers in operation, Y , is recorded for 14 consecutive years in Table P-18. a. Plot number of copy centers versus year. Has On The Double experienced linear or exponential growth? b. Determine the annual growth rate for On The Double. c. Predict the number of copy centers in operation in 2012. Does this number seem reasonable? Why or why not? TABLE P-18 img
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The number of employees ( X ) and profits per employee ( Y ) for n = 16 publishing firms are given in Table P-19. Employees are recorded in thousands and profits per employee in thousands of dollars. A portion of the Minitab output from a straight-line regression analysis is given below. TABLE P-19 img *Dun and Bradstreet. img a. Identify the least squares estimates of the slope and intercept coefficients. b. Test the hypothesis H 0 : ? 1 = 0 with ? =.10. Does there appear to be a relationship between profits per employee and number of employees? Explain. c. Identify and interpret r 2. d. Is this fitted regression function likely to be a good tool for forecasting profits per employee for a given number of publishing firm employees? Discuss.
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Refer to Problem 19. Observation 12 corresponds to Dun and Bradstreet. Redo the straight-line regression analysis omitting this observation. Do your conclusions in parts b and d of Problem 19 change? What, if anything, does this imply about the influence of a single observation on a regression analysis when the number of observations is fairly small? Do you think it is reasonable to throw out Dun and Bradstreet? Discuss.
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