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(A)* Using the following data, construct a simple linear model in R using the variable X as the predictor, and the variable Y as the response.

X <- c(12.47, 8.12, 3.74, 14.07, 9.93, 14.07, 7.18, 10.80, 11.64, 10.69, 11.18, 15.44, 10.90, 11.30, 2.26, 9.95, 17.00, 20.13, 9.35, 10.16, 11.20, 12.09, 10.44, 3.25, 7.25, 2.16, 11.32, 7.17, 6.38, 15.27, 9.94, 10.73, 7.31, 11.18, 16.48, 14.41, 12.42, 12.55, 14.77, 10.19, 12.71, 14.42, 9.39, 12.43, 12.50, 6.55, 6.58, 10.14, 9.21, 10.15, 18.33, 8.15, 15.12, 7.29, 10.69, 6.34, 11.93, 5.64, 11.57, 18.62, 11.09, 0.60, 11.50, 0.81, 11.75, 5.51, 11.14, 13.54, 14.57, 12.72, 6.55, 2.52, 12.83, 7.73, 3.81, 16.71, 5.79, 10.13, 4.32, 15.92, 10.69, 13.21, 12.13, 13.59, 2.32, 16.62, 16.89, 19.51, 7.41, 10.72, 0.65, 13.18, 9.51, 13.02, 9.20, 5.58, 20.28, 5.60, 8.62, 12.42)

Y <- c(50.40, 12.30, 2.68, 38.43, -1.03, 15.52, 45.10, 38.22, 18.60, 26.76, -8.92, 24.48, 35.20, 33.17, 13.86, 32.27, 49.43, 27.53, 10.00, 24.36, 28.80, 9.15, 34.08, 11.64, 19.84, 17.67, 56.08, 24.13, -1.50, 44.86, 39.04, 39.95, 26.21, 6.37, 19.81, 23.36, -5.30, 19.47, 13.34, 12.24, 25.43, 20.05, 32.91, 46.62, 39.62, -0.03, 16.18, 19.60, 19.06, 4.79, 41.86, 11.21, 51.61, 19.20, 12.23, 40.43, 20.67, -8.01, 5.37, 43.91, 19.03, 3.31, 31.11, 35.60, 54.13, -5.03, 5.26, 24.54, 11.15, 10.18, -3.14, 16.77, 16.90, 29.53, 15.11, -7.19, 7.84, 49.21, -3.43, 34.15, 11.30, 23.69, 36.05, 9.34, 17.14, 19.61, 18.04, 45.11, -2.48, 14.68, 5.90, 37.55, 5.87, 9.08, 15.03, -3.52, -13.38, 13.28, -2.34, 1.14) (1) What are the model assumptions being made when simple linear regression analysis is applied to this problem?

(2) What is the coefficient of determination value of the model?

(3) What is the Root Mean Squared Error (RMSE) estimate for the standard error of the model?

(4) What are the estimates for the slope and intercept?

(5) Are the slope and/or intercept found to be statistically significant at a 5% significance level?

(6) Construct a 90% confidence interval for the true mean of the response when the predictor is equal to 10.

(7) Construct an 80% prediction interval for a new response observation when the predictor is equal to 10.

B. Data on structurally deficient highway bridges is compiled by the Federal Highway Administration (FHWA) and reported in the National Bridge Inventory (NBI). For each state, the NBI lists the number of structurally deficient bridges and the total area (thousands of square feet) of the deficient bridges. For future planning and budgeting, the FHWA wants to estimate the total area of structurally deficient bridges in a state based on the number of deficient bridges.

(1) Write the equation of a straight-line model relating total area (y) to number of structurally deficient bridges (x).

(2) The model in part (a) was fit to the data using R as shown in the printout below. ############################

Call: lm(formula = totalarea NumberSDB)

Coefficients:

Estimate Std. Error t value Pr(>|t|)

(Intercept). 119.9 123.0 0.970 0.335

NumberSDB 0.3456. 0.0616 5.61 1.04e-06 ***

Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 635.187 on 48 degrees of freedom

Multiple R-squared: 0.3865, Adjusted R-squared: 0.3743

F-statistic: 31.47 on 1 and 48 DF, p-value: 1.04e-06

############################

Find the least squares prediction equation on the printout.

(3) List the assumptions required for the regression analysis.

(4) Locate the estimated standard error of the regression model, s, on the printout. (e) Give a practical interpretation of the value of s.

(C) The current workforce is dominated by "Generation y"-people born between 1982 and 1999. These workers have a reputation as having an entitlement mentality (e.g., they believe they have a right to a high-paying job without a work ethic). The reasons behind this phenomenon were investigated in Proceedings of the Academy of Educational Leadership (Vol. 16, 2011). A sample of 272 undergraduate business students was administered a questionnaire to capture the behaviors that lead to an entitlement mentality.

The responses were used to measure the following two quantitative variables for each student: entitlement score (y)-where higher scores indicate a greater level of entitlement- and "helicopter parents" score (x)-where higher scores indicate that the student's parents had a higher level of involvement in his or her everyday experiences and problems.

(1) Give the equation of a simple linear regression model relating y to x.

(2) The researchers theorize that helicopter parents lead to an entitlement mentality. Based on this theory, would you expect 0 to be positive or negative (or are you unsure)? Would you expect 1 to be positive or negative (or are you unsure)? Explain.

(3) The p-value for testing H0 : 1 = 0 versus Ha : 1 > 0 was reported as 0.002. Use this result to test the researchers' entitlement theory at = 0.01.

D.* The Chemical Engineering Progress (January 2005) study of the rate at which a spilled volatile liquid spread across a surface. Recall the simple linear regression was used to model y = mass of the spill as a function of x = elapsed time of the spill.

(a) Find a 90% confidence interval for the mean mass of all spills with an elapsed time of 15 minutes. Interpret the result.

(b) Find a 90% prediction interval for the mass of a single spill with an elapsed time of 15 minutes. Interpret the result.