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Statistics For Business And Economics
Found in: Page 807
Statistics For Business And Economics

Statistics For Business And Economics

Book edition 13th
Author(s) James T. McClave, P. George Benson, Terry Sincich
Pages 888 pages
ISBN 9780134506593

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Short Answer

When a multiple regression model is used for estimating the mean of the dependent variable and for predicting a new value of y, which will be narrower—the confidence interval for the mean or the prediction interval for the new y-value? Why?

Confidence interval is narrower than the prediction interval because Prediction intervals must account for both the uncertainty in estimating the population mean, plus the random variation of the individual values. So a prediction interval is always wider than a confidence interval. Also, the prediction interval will not converge to a single value as the sample size increases.

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TABLE OF CONTENTS :
TABLE OF CONTENTS

Difference in confidence and prediction interval

The prediction interval predicts in what range a future individual observation will fall, while a confidence interval shows the likely range of values associated with some statistical parameter of the data, such as the population mean.

Narrower of the two

Confidence interval is narrower than the prediction interval because Prediction intervals must account for both the uncertainty in estimating the population mean, plus the random variation of the individual values. So a prediction interval is always wider than a confidence interval. Also, the prediction interval will not converge to a single value as the sample size increases.

Most popular questions for Math Textbooks

Question: Accuracy of software effort estimates. Periodically, software engineers must provide estimates of their effort in developing new software. In the Journal of Empirical Software Engineering (Vol. 9, 2004), multiple regression was used to predict the accuracy of these effort estimates. The dependent variable, defined as the relative error in estimating effort, y = (Actual effort - Estimated effort)/ (Actual effort) was determined for each in a sample of n = 49 software development tasks. Eight independent variables were evaluated as potential predictors of relative error using stepwise regression. Each of these was formulated as a dummy variable, as shown in the table.

Company role of estimator: x1 = 1 if developer, 0 if project leader

Task complexity: x2 = 1 if low, 0 if medium/high

Contract type: x3 = 1 if fixed price, 0 if hourly rate

Customer importance: x4 = 1 if high, 0 if low/medium

Customer priority: x5 = 1 if time of delivery, 0 if cost or quality

Level of knowledge: x6 = 1 if high, 0 if low/medium

Participation: x7 = 1 if estimator participates in work, 0 if not

Previous accuracy: x8 = 1 if more than 20% accurate, 0 if less than 20% accurate

a. In step 1 of the stepwise regression, how many different one-variable models are fit to the data?

b. In step 1, the variable x1 is selected as the best one- variable predictor. How is this determined?

c. In step 2 of the stepwise regression, how many different two-variable models (where x1 is one of the variables) are fit to the data?

d. The only two variables selected for entry into the stepwise regression model were x1 and x8. The stepwise regression yielded the following prediction equation:

Give a practical interpretation of the β estimates multiplied by x1 and x8.

e) Why should a researcher be wary of using the model, part d, as the final model for predicting effort (y)?

Write a model that relates E(y) to two independent variables—one quantitative and one qualitative at four levels. Construct a model that allows the associated response curves to be second-order but does not allow for interaction between the two independent variables.

Catalytic converters in cars. A quadratic model was applied to motor vehicle toxic emissions data collected in Mexico City (Environmental Science & Engineering, Sept. 1, 2000). The following equation was used to predict the percentage (y) of motor vehicles without catalytic converters in the Mexico City fleet for a given year (x): β^2

a. Explain why the value β^0=325790 has no practical interpretation.

b. Explain why the value β^1=-321.67 should not be Interpreted as a slope.

c. Examine the value of β^2 to determine the nature of the curvature (upward or downward) in the sample data.

d. The researchers used the model to estimate “that just after the year 2021 the fleet of cars with catalytic converters will completely disappear.” Comment on the danger of using the model to predict y in the year 2021. (Note: The model was fit to data collected between 1984 and 1999.)

Question: Revenues of popular movies. The Internet Movie Database (www.imdb.com) monitors the gross revenues for all major motion pictures. The table on the next page gives both the domestic (United States and Canada) and international gross revenues for a sample of 25 popular movies.

  1. Write a first-order model for foreign gross revenues (y) as a function of domestic gross revenues (x).
  2. Write a second-order model for international gross revenues y as a function of domestic gross revenues x.
  3. Construct a scatterplot for these data. Which of the models from parts a and b appears to be the better choice for explaining the variation in foreign gross revenues?
  4. Fit the model of part b to the data and investigate its usefulness. Is there evidence of a curvilinear relationship between international and domestic gross revenues? Try using α=0.05.
  5. Based on your analysis in part d, which of the models from parts a and b better explains the variation in international gross revenues? Compare your answer with your preliminary conclusion from part c.

Question: Novelty of a vacation destination. Many tourists choose a vacation destination based on the newness or uniqueness (i.e., the novelty) of the itinerary. The relationship between novelty and vacationing golfers’ demographics was investigated in the Annals of Tourism Research (Vol. 29, 2002). Data were obtained from a mail survey of 393 golf vacationers to a large coastal resort in the south-eastern United States. Several measures of novelty level (on a numerical scale) were obtained for each vacationer, including “change from routine,” “thrill,” “boredom-alleviation,” and “surprise.” The researcher employed four independent variables in a regression model to predict each of the novelty measures. The independent variables were x1 = number of rounds of golf per year, x2 = total number of golf vacations taken, x3 = number of years played golf, and x4 = average golf score.

  1. Give the hypothesized equation of a first-order model for y = change from routine.

  1. A test of H0: β3 = 0 versus Ha: β3< 0 yielded a p-value of .005. Interpret this result if α = .01.

  1. The estimate of β3 was found to be negative. Based on this result (and the result of part b), the researcher concluded that “those who have played golf for more years are less apt to seek change from their normal routine in their golf vacations.” Do you agree with this statement? Explain.

  1. The regression results for three dependent novelty measures, based on data collected for n = 393 golf vacationers, are summarized in the table below. Give the null hypothesis for testing the overall adequacy of the first-order regression model.

  1. Give the rejection region for the test, part d, for α = .01.

  1. Use the test statistics reported in the table and the rejection region from part e to conduct the test for each of the dependent measures of novelty.

  1. Verify that the p-values reported in the table support your conclusions in part f.

  1. Interpret the values of R2 reported in the table.
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