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. (a) A 90% confidence interval for the mean margin of victory for all presidents with an approval rating of 50%. Round your answers to two decimal places. (b) A 90% prediction interval for the margin of victory for a president with an approval rating of 50%. Round your answers to two decimal places. (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.
Inverse Normal Distribution
The method used for finding the corresponding z-critical value in a normal distribution using the known probability is said to be an inverse normal distribution. The inverse normal distribution is a continuous probability distribution with a family of two parameters.
Mean, Median, Mode
It is a descriptive summary of a data set. It can be defined by using some of the measures. The central tendencies do not provide information regarding individual data from the dataset. However, they give a summary of the data set. The central tendency or measure of central tendency is a central or typical value for a probability distribution.
Z-Scores
A z-score is a unit of measurement used in statistics to describe the position of a raw score in terms of its distance from the mean, measured with reference to standard deviation from the mean. Z-scores are useful in statistics because they allow comparison between two scores that belong to different normal distributions.
4. Predicting Re-Election Margin
Data 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:
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Use values from this output to calculate the following.
(a) A 90% confidence interval for the mean margin of victory for all presidents with an approval rating of 50%. Round your answers to two decimal places.
(b) A 90% prediction interval for the margin of victory for a president with an approval rating of 50%. Round your answers to two decimal places.
(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.
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