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Chapter 12: Q135SE (page 807)

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鈥攖he confidence interval for the mean or the prediction interval for the new y-value? Why?

Short Answer

Expert verified

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.

Step by step solution

01

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.

02

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.

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Most popular questions from this chapter

Going for it on fourth down in the NFL. Refer to the Chance (Winter 2009) study of fourth-down decisions by coaches in the National Football League (NFL), Exercise 11.69 (p. 679). Recall that statisticians at California State University, Northridge, fit a straight-line model for predicting the number of points scored (y) by a team that has a first-down with a given number of yards (x) from the opposing goal line. A second model fit to data collected on five NFL teams from a recent season was the quadratic regression model, E(y)=尾0+尾1x+尾2x2.The regression yielded the following results: y=6.13+0.141x-0.0009x2,R2=0.226.

a) If possible, give a practical interpretation of each of the b estimates in the model.

b) Give a practical interpretation of the coefficient of determination,R2.

c) In Exercise 11.63, the coefficient of correlation for the straight-line model was reported asR2=0.18. Does this statistic alone indicate that the quadratic model is a better fit than the straight-line model? Explain.

d) What test of hypothesis would you conduct to determine if the quadratic model is a better fit than the straight-line model?

Question: Consumer behavior while waiting in line. The Journal of Consumer Research (November 2003) published a study of consumer behavior while waiting in a queue. A sample of n = 148 college students was asked to imagine that they were waiting in line at a post office to mail a package and that the estimated waiting time is 10 minutes or less. After a 10-minute wait, students were asked about their level of negative feelings (annoyed, anxious) on a scale of 1 (strongly disagree) to 9 (strongly agree). Before answering, however, the students were informed about how many people were ahead of them and behind them in the line. The researchers used regression to relate negative feelings score (y) to number ahead in line (x1) and number behind in line (x2).

a.The researchers fit an interaction model to the data. Write the hypothesized equation of this model.

b. In the words of the problem, explain what it means to say that 鈥渪1 and x2 interact to affect y.鈥�

c. A t-test for the interaction 尾 in the model resulted in a p-value greater than 0.25. Interpret this result.

d. From their analysis, the researchers concluded that 鈥渢he greater the number of people ahead, the higher the negative feeling score鈥� and 鈥渢he greater the number of people behind, the lower the negative feeling score.鈥� Use this information to determine the signs of 尾1 and 尾2 in the model.

Suppose the mean value E(y) of a response y is related to the quantitative independent variables x1and x2

E(y)=2+x1-3x2-x1x2

a) Identify and interpret the slope forx2

b) Plot the linear relationship between E(y) andx2for role="math" localid="1649796003444" x1=0,1,2, whererole="math" localid="1649796025582" 1鈮�x2鈮�3

c) How would you interpret the estimated slopes?

d) Use the lines you plotted in part b to determine the changes in E(y) for eachrole="math" localid="1649796051071" x1=0,1,2.

e) Use your graph from part b to determine how much E(y) changes whenrole="math" localid="1649796075921" 3鈮�x1鈮�5androle="math" localid="1649796084395" 1鈮�x2鈮�3.

Consider the following data that fit the quadratic modelE(y)=尾0+尾1x+尾2x2:

a. Construct a scatterplot for this data. Give the prediction equation and calculate R2based on the model above.

b. Interpret the value ofR2.

c. Justify whether the overall model is significant at the 1% significance level if the data result into a p-value of 0.000514.

Production technologies, terroir, and quality of Bordeaux wine. In addition to state-of-the-art technologies, the production of quality wine is strongly influenced by the natural endowments of the grape-growing region鈥攃alled the 鈥渢erroir.鈥� The Economic Journal (May 2008) published an empirical study of the factors that yield a quality Bordeaux wine. A quantitative measure of wine quality (y) was modeled as a function of several qualitative independent variables, including grape-picking method (manual or automated), soil type (clay, gravel, or sand), and slope orientation (east, south, west, southeast, or southwest).

  1. Create the appropriate dummy variables for each of the qualitative independent variables.
  2. Write a model for wine quality (y) as a function of grape-picking method. Interpret the尾鈥檚 in the model.
  3. Write a model for wine quality (y) as a function of soil type. Interpret the尾鈥檚 in the model.
  4. Write a model for wine quality (y) as a function of slope orientation. Interpret the尾鈥檚 in the model.
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