Question 1 A multiple regression was performed of y = life expectancy at birth on X1 = mean years of schooling of adults over the age of 25, X2 = GDP per capita (gross domestic product in constant 2005 international $), and X3 = carbon dioxide emissions per capita (tonnes) for 31 countries. The data were obtained from the human development website http://hdr.undp.org/en/. Based on the following ANOVA table, calculate Source DF SS MS P-Value Regression 3 498.98 166.33 24.16 0.000 Residual (Error) 27 185.88 6.88 Total 30 684.86 0 76.3% O 69.8% O 73.7% O Cannot be determined from the information given.Question 2 Based on the information in the ANOVA table, how many independent variables are in this regression? Source DF SS MS F P-Value Regression 3 14.001 4.67 0.484 0.695 Residual (Error) 41 395.629 9.649 Total 44 409.630 0 3 O 41 O 44 O 2Question 3 Consider the following scatter plot. What can you conclude from this graph? G The dependent and independent variable have a linear relationship 0 The dependent and independent variable have a non-linear relationship Q The dependent and independent variable have a positive linear relationship (3 The dependent and independent variable have a negative linear relationship Question 4 Consider the following normal probability (Q-Q) plot. Normal Probability Plot 250000 200000 150000 100000 50000 20 40 60 80 100 120 Sample Percentile What would you conclude about this data? O This data is perfectly normal O The data appears to be slightly askew from normal O The data is linear O The data follows a Poisson distributionQuestion 5 Does the following residual plot indicate any violation of the regression assumptions? If so, which one? Residual Predicted y values O Zero-mean O Constant variance O Independence O No violationSet up 1. Are more selective colleges more expensive? This is a question asked by students enrolled in 3 statistics course at a liberal arts college. To answer this question, the students collected information on a random sample of41 liberal arts colleges across the nation.They considered the regression model given below. 1.1:": : so + 3, PRIVATE + a: ADMJMTE Variable Description COST Average cost of attendance PRIVATE is the school private? (0 = no, 1 2 es) ADM_RATE Admissions rate Below is the output the students obtained from t1eir favorite statistical software package. LTerm Estimate Std. Error .tvalue Intercept 33670.89 5486.8? 6. l3?r PRIVATE 22713.66 3566.25 6369 ADNLRATE - | 86.60 68.73 9.7 l 5 a\" = 0.5009 Adj: = 0.5799 3 = 8864 What is the interpretation of the estimated slope associated with PRIVATE? For each l-unit increase in private. the average cost of attendance increases by $22,713.66, holding the admissions rate constant. '3 The average cost of attendance fora private school is $22,713.66 higher than the average cost for a public school. assuming that they have the same admissions rate. -.'2 Private schools are $22,713.66 more expensive. on average. '_: None of the answers is correct. Question 7 Use Set up 1. A plot of the residuals versus fitted values is given below. What does this plot reveal about the appropriateness of the assumptions necessary for inference? 10000 - Residual -10000- -20000- 25 50 75 100 Admissions rate O The plot is not useful since it just looks like random scatter. O The spread of the residuals is not constant across admissions rate. O The spread of the residuals is constant across admissions rate. O There are many troubling outliers.Question 8 Use Set up 1. If all assumptions necessary for inference are valid and we wished to determine the overall utility of the model, what hypotheses should we test? O Ho: B2 = 0 versus H, : B2 # 0 O Ho: B, = 0 Versus H,: B, > 0 O Ho: B, = B2 = 0 versus H, : at least one , # 0 Ho: B, = B2 = 0 versus H,: at least one , > 0You are interested in starting a specialty coffee shop, but you need to establish how the volume of production will affect your average total costs before you can complete your business plan. You have a friend with connections in the industry, and she obtains the average total cost per cup for a random sample of several establishments in your region. The final data set your friend compiles contains information on 61 coffee shops including the average total cost of production (in cents per cup) and the rate of output (in cups per hour). The regression output for this simple linear regression model is given below. 150 - 125 100 Average Total Cost (cents/cup) 75 50 60 70 80 90 Output Rate (cups/hr) Source DE SS MS P-Value Regression 1 7003.4 17003.4 118.13 0.0000 Residual (Error) 59 8492.2 143.9 Total 50 25495.6 61 K = Varaiable Estiamte Std. Error t-value p-value Inetercept 181.1443 7.9652 22.74 0.0000 OutputRate -1.4869 0.1368 -10.87 0.0000 Approximately what proportion of the variation in total cost does the regression model explain? O 0.3331 O 0.4994 O 0.6613 O 0.6669Question 10 Below is the graph of the standardized residuals versus one of the independent variables. What do you conclude from this picture? Figure HW15-5.3.2: Plot of Standardized Residuals v. Monthly income of riders (Natural Log] (Data in Natural Logs-Model 2) 25 weekly riders) + Standard Deviation of Residuals Standardized Residuals= (Actual Number of weekly riders. Predicted Number of .. . 9.40 Source: Table WW35-2 (Tab 1) and Table WW25-5.1.2 (Tab 4) Monthly income of riders (Natural Log) O The residuals appear to be getting larger with the size of the variable. This suggests that the variance of the prediction distribution is not constant. O The residuals appear to be getting larger with the size of the variable. This suggests that the variable is a good predictor of the number of trips per week. O The residuals appear to be getting larger with the size of the variable. This suggests that the variance of the prediction distribution is constant. O The residuals appear to be getting larger with the size of the variable. This suggests that the model is linear.Question 11 Below is the QQ plot for residuals. What does this tell us? vmmxuuzmmumwmummmdn nu: wmnlla-IMMI mal mnnnh I] lull-Inu- 0 It is a method to check if the residuals {errors} are normal. 0 It is a method to check if the Y-variable is signicant 0 It can he used to predict the y-variable given an x variable value. 0 It can be used to calculate the slope of the regression line. Question 12 Using Figure 2, what might the above plot be useful for? 8 O O O 03 o To :90,200 Predicted Number of weekly riders O Detecting heteroscedasticity O Testing the null hypothesis O Rejecting the null hypothesis O Detecting multicollinearityQuestion 13 Using Figure 2, one pattern that might be of concern in this graph is O Random pattern of residuals O A megaphone shape O Means of the residuals equaling zero O No patternQuestion 14 Excel exercise. How is the residual (or error) calculated in Model2? O The mean of the Y variable minus its predicted value O The actual Y value minus the predicted Y value. O The sum of the Y value minus the predicted Y value squared O The mean of the first X variable minus the mean of the Y variableQuestion 17 Excel exercise. Consider the following graph, what do you conclude from this picture? Scatter Plot of Residuals v. Predicted Price (Model 2) Standard Residuals 10032 15808 25800 48008 Predicted Price O No pattern is apparent; conclusion is that the variances are all the same O Appears residuals tend to get larger (both positive and negative, as predicted value increases; conclusion is heteroscedasticity. O No pattern is apparent; conclusion is that the variance of the predicted values is the same across values of X. O The pattern is a straight line along a diagonal; conclusion is that the assumption of linearity holds.Question 18 Excel exercise. Consider the below histogram for the standardized residuals. What do you conclude? Histogram of Residuals (Model 2) 221 Frequency 12 81 5.8 -1.78 to - -1 14 to- -0.50 to 0.14 to 0.78 to 142 to 206 to 2.70to 3.34 to 3.98 to 1.14 0.50 0.14 0.78 142 2.06 2.70 3.34 3.98 4.62 O Clearly normally distributed O Skewed to the right O Somewhat normal with skewing to the left O Uniform distribution