In Geographical Analysis, geography professors at the University of Wisconsin-Mi

Question

In Geographical Analysis, geography professors at the University of Wisconsin-Milwaukee and Ohio State University demonstrated the use of satellite image maps for estimating urban population. A proportion of Columbus, Ohio, was partitioned into census block groups, and satellite imagery was obtained. For each census block, the following variables were measured: population density (y), proportion of block with low-density residential areas (x_1), and proportion of block with high-density residential areas (x_2). A multiple regression model for y was fit to the data with the following results: y^cap = -.0304 + 2.006 x_1 + 5.006 x_2, R^2 = .686, R_2^a = .656 a. Give a practical interpretation of R^2 We are 68.6% confident that proportion of blocks with high-density residential areas and proportion of blocks with low-density residential areas are good predictors of population density. Since R^2 = 0.686, r = 0.828. 68.6% of the variability in y can be explained by x_1 68.6% of the variability in y can be explained by the model b. State the null and alternative hypothesis to test if this is a useful model. H_0: beta_1 = beta_2 = 0 H_A: at least one is non-zero H_0: beta_1 = 0 H_A: beta_1 notequalto 0 H_0: beta_2 = 0 H_A: beta_2 notequalto 0 c. What test statistic would you use to conduct this test Chi-Square F T Z d. The test in (b.) was conducted and the result was a p-value of 0.004. If alpha = 0.01, would you conclude that this model is useful? No, p-value is less than alpha, Yes, p-value is less than alpha. e. What value would you use to compare a model with two predictors for y to model with three predictors for y, if both models were found to be effective? T statistic r R^2 R_2^a

Explanation / Answer

R-squared is a statistical measure of how close the data are to the fitted regression

The adjusted R-squared is a modified version of R-squared

A)

R-squared is a statistical measure of how close the data are to the fitted regression

Hence in the example 68.6% of the variability in Y can be explained by the model Right Answer : iv. 68.6% of the variability in Y can be explained by the model B) Since we have to test the dependence of population density (y) on both the independent variables viz proportion of block with low-density residential areas (x1) and proportion of block with high density residential areas (x2), the appropriate null and alternative hypothsis are option (i) Right Answer : Ho : 1 = 2 = 0 Ha : atleast one is non-zero C) We use the F-statistic for conducting the test Right Answer : ii. F D) p-value = 0.004 = 0.01 Since p-value is less than , we reject the null hypothesis This means that the model gives a better fit Right Answer :     ii.   Yes, p-value is less than E)

The adjusted R-squared is a modified version of R-squared

that has been adjusted for the number of predictors in the model. Hence for comparing models with 2 predictors and 3 predictors, we will use the R2 adjusted Right Answer :     iv. R2a

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