Filling in a simple linear regression ANOVA table (You will be marked on: 1) Getting the proper R output and plots, 2) Validity of your statistical conclusions and interpretations, 3) Writing style (grammar and clear concise language count!), 4) Presentation (I'm not planning on being super picky on this point, but please make your submission easy to follow).

Suppose we have a simple linear regression setting, where we have sampled 100 (X, Y) pairs of values. When running a simple linear regression on this data, we find that 12% of the variance in Y can be explained by the linear relationship with X. If we completely ignore X, and calculate the sample variance of the Y values, we find that it's 50.0. Complete the simple linear regression ANOVA table (the full table, as seen in class). Use R to find the p-value, and include that as well.

Example 1 Researchers investigated a possible relationship between lean body mass and metabolic rate (Calories/24 hour) on 12 female volunteers. Mass 36.1 54.6 48.5 42.0 50.6 42.0 40.3 33.1 42.4 34.5 51.1 41.2 Metabolic rate 995 1425 1396 1418 1502 1256 1189 913 1124 1052 1347 1204 1500 1300 Metabolic Rate 1100 900 35 40 45 50 55 Mass (kg)4 Analysis of Variance (ANOVA) in Regression In the regression analyses that we will carry out, ANOVA will help us test the null hypothesis that one or more slopes equal 0. ANOVA for regression operates on the principle of the partitioning of the sums of squares: [ (vi - n) = [ m - ) + [(x-x)? The total sum of squares (E(Y; - Y)?) is partitioned into the sum of squares for regression (E(Y - Y)2) and the sum of squares for residual (E(Y; - Y;)?). (This identity is not obvious, but it is not too hard to show algebraically.) It is also true that the degrees of freedom for regression and residual sum to the degrees of freedom for residual (error): DF(Total) = DF(Regression) + DF(Residual) The results are illustrated in an ANOVA table: Source Degrees of freedom Sum of Squares Mean Square F Regression E( Yi - Y)2 MSReg = SSReg/ 1 MSReg MSRes Residual n 2 E(Yi - Y:)2 MSRes = SSRes/ (n - 2) Total E(Y - Y)2