Reference no: EM132263569

Homework -

Q1. In a small-scale experimental study of the relation between degree of brand preference (Y), moisture content (X₁), and sweetness (X₂) of the product, results were obtained from an experiment based on a completely randomized design. In what follows, you are to treat Y as the dependent variable and X₁, X₂, and any functions of X₁ and X₂ as independent variables. The following full model was fit to the data

Y_i = β₀ + β₁X_1i + β₂X_2i + β₃(X_1i ? X_2i) + β₄X_1i² + β₅X_2i² + ε_i,

and the corresponding R output is given below (in attached file). Assume α = 0.05 for all tests.

a) Calculate the sample variance for Y (i.e. S_Y²).

b) What is the value of R²?

c) Consider the following three sets of hypotheses: H₀ : β₃ = 0 versus H₁ : β₃ ≠ 0, H₀ : β₄ = 0 versus H₁ : β₄ ≠ 0, and H₀ : β₅ = 0 versus H₁ : β₅ ≠ 0. Give the test statistic values and corresponding p-values (in R, use the pt function) for each test. What are your conclusions for each test?

d) Consider testing H₀ : β₃ = β₄ = β₅ = 0 versus H₁ : β₃ ≠ 0, β₄ ≠ 0, and/or β₅ ≠ 0. The appropriate reduced model was fit and the corresponding R output is given below. Use this output to calculate the (F) test statistic value and corresponding p-value (in R, use the pf function) and state your conclusion?

e) What is the fundamental difference between parts c and d? Based on parts c and d, do you think that the "curvature" terms (i.e. X₁ ? X₂, X₁², and X₂²) are important in explaining brand preference?

Q2. 93 cars were selected at random from among passenger car models. The data (which can be found on E-Learning under Content - Data Sets - cars) includes the following variables: HP: Horsepower (maximum), C: Number of cylinders, E: Engine size (liters) and W: Weight (pounds). For this problem, you are to treat HP as the dependent (or response) variable and C, E, and W as the independent (or predictor) variables. Furthermore, assume α = 0.05 for all tests.

a) Fit a multiple linear regression model relating horsepower (HP) to number of cylinders (C), engine size (E), and weight (W). What is the F-statistic and corresponding p-value for the test of "regression significance"?

b) What are the values of R² and R_adj². Interpret the value of R² for this problem.

c) Obtain a 95% confidence interval for the slope parameter pertaining to weight (W). You can use the user-defined betaci function.

d) Obtain a 95% confidence interval for the mean horsepower response when C = 3, E = 2.5, and W = 3000.

e) Obtain a 95% prediction interval for a new observation of horsepower when C = 2, E = 3.0, and W = 2800.

Q3. The file (the data can be found on E-Learning under Content - Data Sets - airpass) airpass.txt contains a single column of data. This data represents a so-called time series that contains the number of international airline passengers (in thousands) for each month from January, 1949 through December, 1960. Thus, n = 144 observations. In what follows, you are to treat the logarithm of this data as the dependent (or response) variable (i.e. in R, use Y = log(airpass) to create the variable).

a) Let denote the independent variable "time". That is, takes on the values 1, 2, 3, ..., 144 corresponding to the first, second, third, ..., and last values of , respectively (In R, use x1 = 1:144). Obtain a time series plot of the data using the following command: plot(x1,y,type="l"). Comment on any trends and/or periodic features you see in this plot.

b) Let X₂ = cos(2πX₁/12) and X3 = sin(2πX₁/12) be two additional independent variables (e.g. In R, use x2 = cos((2*pi*x1)/12) and x3 = sin((2*pi*x1)/12)) and fit the following model to the data: Yi = β₀ + β₁X_1i + β₂X_2i + β₃X_3i + ε_i. What is the value of R² for this model?

c) With the help of the predict command, obtain the model (given in (b)) predictions when X₁ = 1.00, 1.01, 1.02, 1.03, ..., 143.98, 143.99, 144.00 and store these predictions in a vector called "Z". What are the model predictions when X₁ = 1.00 and X₁ = 144.00, respectively?

d) Referring back to parts (a) and (c), add a red prediction line to the plot in part (a) with a command similar to the following: lines(seq(1,144,.01),Z,col=2). As part of the output to this question, include this plot with your solution. By visual inspection of this plot, how well do you think the model did in capturing the unique features of this data?

e) Consider testing H₀ : β₂ = β₃ = 0 versus H₁ : β₂ ≠ 0 and/or β₃ ≠ 0? Obtain the corresponding F-statistic and p-value for this test using the user-defined betatest function. What is your conclusion? Also, how does this conclusion relate to your interpretation of the plot given in part (d)?

f) Back in December, 1960, what would have been a 95% prediction interval for the log of the number of international airline passengers for January, 1961?

4. A random sample of 32 births were collected by researchers to investigate if smoking mothers have babies with lower birth weight? The researchers collected data on birth weight (Weight), length of gestation (Gest), and smoking status of the mother (smoker or non-smoker). The data can be found on E-Learning under Content - Data Sets - birthweight. In what follows, let Y = Weight, X1 be a 0/1 indicator variable (i.e. 0 = non-smoker, 1 = smoker) that represents the independent variable "smoker", and X2 = Gest. Assume α = 0.05 for all hypothesis testing questions.

a) Fit the following SLR model: Y_i = β₁₀ + β₁₁X1_i + ε_i and answer the following questions.

i. What is the estimate of and what is its interpretation in terms of this problem?

ii. What are the test statistic and p-value for testing H₀ : β₁₁ = 0 versus H₁ : β₁₁ ≠ 0? Based on the corresponding p-value for this model, does being a "smoker" significantly affect the birth weight of babies?

iii. What are the values of R², R_adj², and σ^{^2} for this model?

b) Next, fit the following MLR model: Y_i = β₂₀ + β₂₁X1_i + β₂₂X2_i + ε_i and answer the following questions.

i. What is the estimate β₂₁ of and what is its interpretation in terms of this problem?

ii. What are the test statistic and p-value for testing H₀ : β₂₁ = 0 versus H₁ : β₂₁ ≠ 0? Based on the corresponding p-value for this model, does being a "smoker" significantly affect the birth weight of babies?

iii. What are the values of R², R_adj², and σ^{^2} for this model?

c) Are your overall conclusions for parts (a) and (b) the same? If so, why? If not, why? Hint: Consider the values of σ^{^2} for each model.

Note - You can ignore part e for question 3. The data files have been attached.

Attachment:- Assignment Files.rar