4. Predicting Re-Election MarginData has been introduced on the approval rating of an incumbent US president and the margin of victory or defeat in the subsequent election (where negative numbers indicate the margin by which the incumbent president lost the re-election campaign). The data are reproduced in the table below and are available in ElectionMargin. Approval 62 50 70 67 57 48 31 57 39 55 49 50 Margin 10.0 4.5 15.4 22.6 23.2 -2.1 -9.7 18.2 -5.5 8.5 2.4 3.9 Table 1 Presidential approval rating and margin of victory or defeat Computer output for summary statistics for the two variables and for a regression model to predict the margin of victory or defeat from the approval rating is shown: Variable N Mean StDev Approval 12 52.92 11.04 Margin 12 7.62 10.72 ------------------------- The regression equation is  Margin=-36.76+0.839⁢ Approval Term Coef SE Coef T P Constant -36.76 8.34 -4.41 0.001 Approval 0.839 0.155 5.43 0.000 S=5.66054 R-Sq=74.64% R-Sq(adj)=72.10% Analysis of Variance Source DF SS MS F P Regression 1 943.0 943.04 29.43 0.000 Residual Error 10 320.4 32.04     Total 11 1263.5       Use values from this output to calculate the following.   (c) A 90% confidence interval for the mean margin of victory if we have no information about the approval rating. Round your answers to two decimal places. (Note: the answer is not -51.87 to -21.65)

Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...