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Let \(x\) be the size of a house (in square feet) and \(y\) be the amount of natural gas used (therms) during a specified period. Suppose that for a particular community, \(x\) and \(y\) are related according to the simple linear regression model with \(\beta=\) slope of population regression line \(=.017\) \(\alpha=y\) intercept of population regression line \(=-5.0\) Houses in this community range in size from 1000 to 3000 square feet. a. What is the equation of the population regression line? b. Graph the population regression line by first finding the point on the line corresponding to \(x=1000\) and then the point corresponding to \(x=2000\), and drawing a line through these points. c. What is the mean value of gas usage for houses with 2100 sq. ft. of space? d. What is the average change in usage associated with a 1 sq. ft. increase in size? e. What is the average change in usage associated with a 100 sq. ft. increase in size? f. Would you use the model to predict mean usage for a 500 sq. ft. house? Why or why not?

The paper "Predicting Yolk Height, Yolk Width, Albumen Length, Eggshell Weight, Egg Shape Index, Eggshell Thickness, Egg Surface Area of Japanese Quails Using Various Egg Traits as Regressors" (International journal of Poultry Science [2008]: \(85-88\) ) suggests that the simple linear regression model is reasonable for describing the relationship between \(y=\) eggshell thickness (in micrometers) and \(x=\) egg length (mm) for quail eggs. Suppose that the population regression line is \(y=0.135+0.003 x\) and that \(\sigma=0.005 .\) Then, for a fixed \(x\) value, \(y\) has a normal distribution with mean \(0.135+0.003 x\) and standard deviation 0.005 . a. What is the mean eggshell thickness for quail eggs that are \(15 \mathrm{~mm}\) in length? For quail eggs that are \(17 \mathrm{~mm}\) in length? b. What is the probability that a quail egg with a length of \(15 \mathrm{~mm}\) will have a shell thickness that is greater than \(0.18 \mu \mathrm{m} ?\) c. Approximately what proportion of quail eggs of length \(14 \mathrm{~mm}\) have a shell thickness of greater than \(0.175 ?\) Less than \(0.178 ?\)

Hormone replacement therapy (HRT) is thought to increase the risk of breast cancer. The accompanying data on \(x=\) percent of women using HRT and \(y=\) breast cancer incidence (cases per 100,000 women) for a region in Germany for 5 years appeared in the paper "Decline in Breast Cancer Incidence after Decrease in Utilisation of Hormone Replacement Therapy" (Epidemiology [2008]: \(427-430\) ). The authors of the paper used a simple linear regression model to describe the relationship between HRT use and breast cancer incidence. \begin{tabular}{|cc|} \hline HRT Use & Breast Cancer Incidence \\ \hline 46.30 & 103.3 \\ 40.60 & 105.0 \\ 39.50 & 100.0 \\ 36.60 & 93.8 \\ 30.00 & 83.5 \\ \hline \end{tabular} a. What is the equation of the estimated regression line? b. What is the estimated average change in breast cancer incidence associated with a 1 percentage point increase in HRT use? c. What would you predict the breast cancer incidence to be in a year when HRT use was \(40 \% ?\) d. Should you use this regression model to predict breast cancer incidence for a year when HRT use was \(20 \%\) ? Explain. e. Calculate and interpret the value of \(r^{2}\). f. Calculate and interpret the value of \(s_{e}\).

A journalist is reporting about some research on appropriate amounts of sleep for people 9 to 19 years of age. In that research, a linear regression model is used to describe the relationship between alertness and number of hours of sleep the night before. The researchers reported a \(95 \%\) confidence interval, but newspapers usually report an estimate and a margin of error. Explain how the journalist could determine the margin of error from the reported confidence interval.

Do taller adults make more money? The authors of the paper "Stature and Status: Height, Ability, and Labor Market Outcomes" (Journal of Political Economics [2008]: 499-532) investigated the association between height and earnings. They used the simple linear regression model to describe the relationship between \(x=\) height (in inches) and \(y=\) log(weekly gross earnings in dollars) in a very large sample of men. The logarithm of weekly gross earnings was used because this transformation resulted in a relationship that was approximately linear. The paper reported that the slope of the estimated regression line was \(b=0.023\) and the standard deviation of \(b\) was \(s_{b}=0.004\). Carry out a hypothesis test to decide if there is convincing evidence of a useful linear relationship between height and the logarithm of weekly earnings. You can assume that the basic assumptions of the simple linear regression model are met.

15.19 Acrylamide is a chemical that is sometimes found in cooked starchy foods and which is thought to increase the risk of certain kinds of cancer. The paper "A Statistical Regression Model for the Estimation of Acrylamide Concentrations in French Fries for Excess Lifetime Cancer Risk Assessment" (Food and Chemical Toxicology [2012]: \(3867-3876\) ) describes a study to investigate the effect of frying time (in seconds) and acrylamide concentration (in micrograms per kilogram) in french fries. The data in the accompanying table are approximate values read from a graph that appeared in the paper. \begin{tabular}{|cc|} \hline Frying Time & Acrylamide Concentration \\ \hline 150 & 155 \\ 240 & 120 \\ 240 & 190 \\ 270 & 185 \\ 300 & 140 \\ 300 & 270 \\ \hline \end{tabular} a. For these data, the estimated regression line for predicting \(y=\) acrylamide concentration based on \(x=\) frying time is \(y=87+0.359 x\). What is an estimate of the average change in acrylamide concentration associated with a 1-second increase in frying time? b. What would you predict for acrylamide concentration for a frying time of 250 seconds? c. Use the given Minitab output to decide if there is convincing evidence of a useful linear relationship between acrylamide concentration and frying time. You may assume that the necessary conditions have been met. R-sq \(\begin{array}{cc}\text { R-sq(adj) } & \text { R-sq(pred) } \\ 0.00 \% & 0.00 \%\end{array}\) \(\mathrm{S}\) 3 \(\mathrm{q}\) \(8 \%\) Coefficients \(\mathrm{K}-\mathrm{Sq}\) \(14.38 \%\) \(\begin{array}{lccccc}\text { Term } & \text { Coef } & \text { SE Coef } & \text { T-Value } & \text { P-Value } & \text { VIF } \\ \text { Constant } & 87 & 112 & 0.78 & 0.480 & \\ x & 0.359 & 0.438 & 0.82 & 0.459 & 1.00\end{array}\) Regression Equation \(y=87+0.359 x\)

Consider a test of hypotheses about, \(\beta\) the population slope in a linear regression model. a. If you reject the null hypothesis, \(\beta=0\), what does this mean in terms of a linear relationship between \(x\) and \(y ?\) b. If you fail to reject the null hypothesis, \(\beta=0,\) what does this mean in terms of a linear relationship between \(x\) and \(y ?\)

The article "Vital Dimensions in Volume Perception: Can the Eye Fool the Stomach?" (Journal of Marketing Research [1999]: \(313-326\) ) gave the accompanying data on the dimensions (in \(\mathrm{cm}\) ) of the containers for 27 representative food products (Gerber baby food, Cheez Whiz, Skippy Peanut Butter, and Ahmed's tandoori paste, to name a few). a. Fit the simple linear regression model that would allow prediction of the maximum width of a food container based on its minimum width. b. Calculate the standardized residuals (or just the residuals if you don't have access to a computer program that gives standardized residuals) and make a residual plot to determine whether there are any outliers. c. The data point with the largest residual is for a 1 -liter Coke bottle. Delete this data point and refit the regression. Did deletion of this point result in a large change in the equation of the estimated regression line? d. For the regression line of Part (c), interpret the estimated slope and, if appropriate, the estimated intercept. e. For the data set with the Coke bottle deleted, do you think that the assumptions of the simple linear regression model are reasonable? Give statistical evidence for your answer.

Explain what distinguishes a deterministic model from a probabilistic model.

The SAT and ACT exams are often used to predict a student's first-term college grade point average (GPA). Different formulas are used for different colleges and majors. Suppose that a student is applying to State U with an intended major in civil engineering. Also suppose that for this college and this major, the following model is used to predict first term GPA. $$ \begin{aligned} G P A &=a+b(A C T) \\ a &=0.5 \\ b &=0.1 \end{aligned} $$ a. In this context, what would be the appropriate interpretation of the value of \(a\) ? b. In this context, what would be the appropriate interpretation of the value of \(b ?\)