Statistics Questions

QMB3200 – HW#3– Dummies and curvature – Summer 2021 15 pts – Due sat Jun 19 @ 8am Name: Consumer Reports tested 19 different brands and models of bikes. Road bikes are designed for long trips and fitness bikes are designed for daily commutes and regular workouts. Data available for this set of bikes is comprised of weight of the bike (in pounds) and its price (in $). You are hired as a statistician to analyze how much of the weight and the type of bike characteristics determine its price. Model 1: include only the variable Weight on the linear regression a. (1pt) Comment on the scatter diagram for the variables price and bike weight. Answer: b. (1pt) Comment on the goodness of fit of MODEL 1. Answer: c. (1pt) Report the statistical significance of MODEL 1 Answer: Model 2: include the dummy variable Dtype which equals 1 if the bike type is Road, zero otherwise a. (1pt) Comment on the goodness of fit of MODEL

2. Answer: b. (3pt) Report the statistical significance of the coefficients for MODEL 2 Answer: c. (3pt) Interpret the intercept in this model. Answer: d. (2pt) Interpret the coefficient for the variable Dtype. Answer: Model 3: capture curvature with a quadratic functional form a. (1pt) Curvature from the data is captured by a quadratic model using only Weight. Considering model 1, model 2 and the quadratic model below, which one you think fits the data best? Explain why for full credit. Answer: b. (1pt) Write down the quadratic model equation. Answer: c. (1pt) What is the average price in the quadratic model? Answer: 1 QMB3200 – HW#3– Dummies and curvature – Summer 2021 15 pts – Due sat Jun 19 @ 8am Name: Consumer Reports tested 19 different brands and models of bikes. Road bikes are designed for long trips and fitness bikes are designed for daily commutes and regular workouts. Data available for this set of bikes is comprised of weight of the bike (in pounds) and its price (in $).

You are hired as a statistician to analyze how much of the weight and the type of bike characteristics determine its price. Model 1: include only the variable Weight on the linear regression a. (1pt) Comment on the scatter diagram for the variables price and bike weight. Answer: b. (1pt) Comment on the goodness of fit of MODEL 1. Answer: c. (1pt) Report the statistical significance of MODEL 1 Answer: Model 2: include the dummy variable Dtype which equals 1 if the bike type is Road, zero otherwise a. (1pt) Comment on the goodness of fit of MODEL 2. Answer: b. (3pt) Report the statistical significance of the coefficients for MODEL 2 Answer: c. (3pt) Interpret the intercept in this model. Answer: d. (2pt) Interpret the coefficient for the variable Dtype. Answer: Model 3: capture curvature with a quadratic functional form a. (1pt) Curvature from the data is captured by a quadratic model using only Weight. Considering model 1, model 2 and the quadratic model below, which one you think fits the data best? Explain why for full credit.

Answer: b. (1pt) Write down the quadratic model equation. Answer: c. (1pt) What is the average price in the quadratic model? Answer: 1 QMB3200 – HW#3 – Dummies and curvature – Summer 2021 15 pts – Due Sat Jun 19 @ 8AM Scatter Plot Model 1: include only the variable Weight on the linear regression Model 1 Summary Model R Std. Error of the Square Estimate R Square .715a 1 Adjusted R .511 .484 350.54125 a. Predictors: (Constant), weight ANOVAa Model 1 Sum of Squares df Mean Square Regression 2313671.914 1 2313671.914 Residual 2211825.174 18 122879.176 Total 4525497.088 19 F Sig. 18.829 .000b a. Dependent Variable: Price b. Predictors: (Constant), weight Model 1 Coefficientsa Standardized Unstandardized Coefficients Model 1 B (Constant) weight Std. Error 2598.086 432.397 -72.122 15.689 Coefficients Beta t -.758 Sig. 6.787 .000 -4.929 .002

a. Dependent Variable: Price 1 Model 2: include the dummy variable Dtype which equals 1 if the bike type is Road, zero otherwise Model 2 Summary Model R Std. Error of the Square Estimate R Square .768a 1 Adjusted R .590 .542 322.78254 a. Predictors: (Constant), Dtype, weight Model 2 Coefficientsa Standardized Unstandardized Coefficients Model 1 B (Constant) Coefficients Std. Error 1296.298 809.505 weight -30.535 27.222 Dtype 474.308 256.753 Beta t Sig. 1.136 .036 -.432 -1.696 .288 .369 1.384 .089 a. Dependent Variable: Price Model 3: capture curvature with a quadratic functional form Model 3 Summary and Parameter Estimates Dependent Variable: Price Model Summary Equation Quadratic R Square .768 F 38.732 df1 Parameter Estimates df2 2 Sig. 17 .000 Constant 12057.124 b1 -796.803 b2 13.419 The independent variable is weight. 2

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