Statistics II Exercises Chapter 5

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1 Statistics II Exercises Chapter 5 1. Consider the four datasets provided in the transparencies for Chapter 5 (section 5.1) (a) Check that all four datasets generate exactly the same LS linear regression equation. (b) Apply to dataset # 1 the methods for detecting the presence of specification error discussed (c) Apply to dataset # 2 the methods for detecting the presence of specification error discussed (d) Apply to dataset # 3 the methods for detecting the presence of specification error discussed (e) Find the outlier in dataset # 3. Obtain the LS regression line after eliminating this data point and comment on the results. 2. Using a sample of 30 observations, the following linear regression model was estimated ŷ i = ˆβ 0 + ˆβ 1 x i, with ˆβ 0 = 10.1 and ˆβ 1 = 8.4. The sum of squared deviations from the mean for the response variable, due to the model, is i (ŷ i ȳ) 2 = 128, while the sum of squared residuals is i e2 i = 286. (a) Compute the Coefficient of Determination and interpret it. (b) What can you say regarding the correlation coefficient between the values x i and y i? (c) Build the corresponding ANOVA table using these data. (d) Test, at the 5% significance level, the hypothesis that the response variable y does not depend on x. Repeat this test at the 1% significance level. (e) Provide an unbiased estimator for the variance of the error term. 3. The manager of a car dealership is interested in finding the statistical relationship between the number of salesmen employed during weekends and the corresponding total number of cars sold. The following data were obtained over the course of 6 consecutive weekends: x i (# of salesmen) y i (# of cars sold) (a) Find the LS regression line for y (# of cars sold) as a function of x (# of salesmen). (b) Build the corresponding ANOVA table and check the validity of the decomposition TSS = ESS + RSS. (c) Compute the coefficient of determination and interpret it. (d) Use the ANOVA table to test, at the 1% and 5% significance levels, the hypothesis that the number of salesmen employed during the weekend does not affect the corresponding total number of cars sold. (e) Test the same hypotheses as in the preceding question, but using the procedure studied in Chapter 4. Check that the corresponding test statistic T and the test statistic F you used before satisfy the equality F = T Using the data transformations seen in class, linearize the following nonlinear relationships: (a) y = ln(5 x). 1

2 (b) y = 2 3 8x. (c) y = 1/(4 x). (d) y = 5 4 x. 5. Assume we have obtained the following measurements for a response variable y as a function of the explanatory variable x: x i y i (a) Draw the scatterplot (x i, y i ). Do you think a linear relationship can adequately describe this dataset? (b) Assuming an adequate model is of the form y = ax b u, apply the correct transformations to the variables x and y, and obtain point estimates of the parameters a and b from an LS linear regression model between the transformed variables. (c) Build the ANOVA table for the transformed variables. Obtain and interpret the corresponding coefficient of determination. 6. For dataset # 1 from Exercise 1, obtain the LS estimators for the linear regression coefficients using vector-matrix notation. 7. Using n = 34 observations the following multiple linear regression model was estimated: ŷ = x x 2 7.2x 3. The standard errors of the regression coefficient estimates for the explanatory variables are as follows s( ˆβ 1 ) = 3.1, s( ˆβ 2 ) = 3.7 and s( ˆβ 3 ) = 3.2. The corresponding coefficient of determination is R 2 = (a) Obtain confidence intervals at the 95% level for the regression coefficients corresponding to each explanatory variable. (b) For each explanatory variable, test at the 5% significance level the hypothesis that the response does not depend on that variable. (c) Is there evidence, at the 1% significance level, that for each explanatory variable the corresponding regression coefficient is positive? 8. Assume you have obtained estimates of the regression coefficients for the following model y i = β 0 + β 1 x β k x k + u i. Test, at the 5% significance level, the null hypotheses that the response does not depend on any of the explanatory variables, using the following partial ANOVA tables: (a) (b) (c) Model Residuals Model Residuals Model Residuals For 10 single-family houses we obtained their price (in Me), the size of their built area (in m 2 ), the size of the surrounding terrain (in Has.), and the number of bathrooms: 2

3 price (Me) built area (m 2 ) terrain (Has.) # bathrooms Below we give the Statgraphics output for the linear multiple regression of y (price) over x 1 (built area), x 2 (size of terrain), and x 3 (# of bathrooms). (a) Obtain 95% confidence intervals for the regression coefficients of the linear model y = β 0 + β 1 x 1 + β 2 x 2 + β 3 x 3 + u. (b) Test at the 5% and 10% significance levels the hypotheses that the response variable does not depend on x j, for j = 1, 2, 3. (c) Obtain and interpret the value of the coefficient of determination. Obtain an estimate of the standard deviation of the error term. Multiple Regression Analysis Dependent variable: precio Standard T Parameter Estimate Error Statistic P-Value CONSTANT 100,985 7, ,844 0,0000 superfi 0, , , ,0109 superfterreno 109,115 73,4594 1, ,1880 WCs 10,3945 6, , ,1807 Analysis of Variance Source Sum of Squares Df Mean Square F-Ratio P-Value Model 6158, ,99 73,92 0,0000 Residual 166, ,7726 (Corr.) 6325, For the multiple linear regression model y i = b 0 +b 1 x 1i +b 2 x 2i +u i, where we assume that the usual hypotheses hold, we are given a simple random sample of size 4. The data, of the form (y i, x 1i, x 2i ), are (y 1, 1, 2), (2, 2, 3), (y 3, 3, 5) and (y 4, 1, 3), that is, we do not know the value of the variable y for individuals 1, 3 and 4. We know that the LS estimates of the model parameters are ˆb 0 = 1.7, ˆb 1 = -0.3 and ˆb 2 = 0.1. We also know You are asked to: (X X) 1 = (a) Compute the unknown values y 1, y 3, y (b) Estimate the variances of the estimators for the parameters,. ˆ V ar(ˆb i ) for i {1, 2, 3}. (c) Construct the ANOVA table and test at the 95% level the global significance of the model, that is, test H 0 : b 1 = b 2 = 0. 3

4 (d) For a simple linear regression model y i = b 0 + b 1 x i + u i, where we assume that the usual hypotheses hold, assume also that we are given a simple random sample of size n, composed of pairs (y 1, x 1 ),, (y n, x n ). We can write down the variance of ˆb 1 in two different ways: the first one would be Var(ˆb 1 ) = σ2 ns 2 x, where S 2 x = (1/n) n (x i x) 2, and the second one would be based on the matrix representation of the model, given by corresponding element of the matrix σ 2 (X X) 1. Prove that both representations are equivalent. 11. The Tourism Department of the Madrid regional government wishes to analyze the relationship between the yearly tourism promotion budgets and education and infrastructures budgets, for towns in the region having less than inhabitants. 20 towns were randomly selected for this study, and the following variables were measured: y = yearly tourism promotion budget (in millions of euros). x 1 = yearly education budget (in millions of euros). x 2 = yearly infrastructures budget (in millions of euros). From this study the following information was obtained: (X T X) 1 = X T Y = We also know that the residual sum of squares is and the total sum of squares is For the multiple linear regression model: you are asked to: RSS = (y i ŷ i ) 2 = TSS = (y i ȳ) 2 = 0.1 y i = β 0 + β 1 x i1 + β 2 x i2 + u i i = 1..., 20, (a) Estimate and interpret the coefficients of the linear regression model. (b) Build the ANOVA table for this regression and conduct a global significance test. (c) Conduct significance tests for each of the parameters in the model, indicating in each case the null and alternative hypotheses. What can you comment on the significance of the parameters, when compared to the results for the preceding question? (d) Compute a forecast for the yearly tourism promotion budget for a town with a yearly education budget of 1.3 (millions of euros) and an infrastructure budget of 1.2 (millions of euros). 12. You are given a sample of 20 observations {x, z, y} of values from three variables, X, Y and Z. For this sample you have that yi 2 = 10.08, ȳ = You have computed the LS estimates for the regression model y = β 0 + β 1 x + β 2 z + u. The values you have obtained are: ˆβ 0 = 0.065, ˆβ1 = , ˆβ2 = 0.104, s( ˆβ 1 ) = 0.152, s( ˆβ 2 ) = 0.028, e 2 i = If we accept that the assumptions for the linear regression model hold, answer the following questions: 4

5 (a) Complete the ANOVA table for this multiple linear regression model. (b) Compute the coefficient of multiple determination for the model and comment on its value. (c) Test if the model is globally significant to explain the values of Y as a linear function of X and Z, at a 1% significance level. (d) Test if you would have sufficient evidence to conclude that increases in the value of the variable X imply decreases in the value of the variable Y (while Z remains constant), at a 5% significance level. 5

Estadística II Chapter 5. Regression analysis (second part)

Estadística II Chapter 5. Regression analysis (second part) Chapter 5. Regression analysis (second part) Contents Diagnostic: Residual analysis The ANOVA (ANalysis Of VAriance) decomposition Nonlinear