Question

3. Placing all of the predictor variables in a single model is fine if we just want to make predictions, but in this case we are interested in interpreting the slopes. Having correlated predictors e.g. Expend 19ร9 and Expend1960) in a regression model can cause the standard errors of the slopes to increase dramatically, rendering their traditional interpretations unreliable. It can also be challenging to make sense of what it means to change the value of one variable while holding the values of a large number of other variables constant. For these reasons, we should reduce the number of predictor variables in our model. Suppose we are interested in estimating the differences between mean robbery rates in southern states and non-southern states. We will use Expend1960 and MalesPerFemales as potential covariates. For parts a. through d. below estimate the given model, and enter estimated coefficients for each predictor, along with their corresponding p-values, and the R2 statistic for the model. Note that p-values for the intercepts are not requested, because the intercept is not of scientific interest. (Important: you should roport JMP output to 4 decimal placas. If you don't do this, you may and up gatting subsequent calculations wrong.) Robbery-Rate- Po β1Southerni + si a. bo- 1- b. Robbery Ratei o+ 1Southerni +2MalesPerfemales e p-value- bo- c. Robbery-Rate.-Po + β|Southern + β2Expend1960 + εί bo- d. Robbery-Rate . β0 + β 1Southerni + β2Expend1960 bo- β3MalesPerFemales p-value-

Data is copied below for ease of use.

92   1   123   115   978   31   6310   10.4
75.2   0   128   128   934   51   6270   11.6
121.9   0   113   105   985   78   6260   10.8
74.2   0   74   67   984   34   5570   10.8
44.8   1   47   44   962   22   2880   8.9
120.8   0   87   83   953   43   5130   9.6
96   0   78   73   1038   7   5400   11.6
50.2   0   63   57   984   14   4860   11.6
200   0   160   143   1071   3   6740   12.1
34.5   0   69   71   965   6   5640   10.9
123   0   82   76   1018   10   5370   11.2
104.3   0   166   157   938   168   6370   10.7
70.6   1   58   54   973   46   3960   8.9
37.5   0   55   54   1045   6   4530   9.3
76.6   0   90   81   964   97   6170   10.9
108.3   1   63   64   972   23   4620   10.4
90.6   0   97   97   990   18   5890   11.8
64.8   0   97   87   948   113   5720   10.2
127.4   0   109   98   964   9   5590   10
81.4   1   58   56   974   24   3820   8.7
56.1   0   51   47   1024   7   4250   10.4
82.8   1   61   54   953   36   3950   8.8
115.2   1   82   74   981   96   4880   10.4
88.1   0   72   66   998   9   5900   12.2
53.8   0   56   54   968   4   4890   10.9
81.8   1   75   70   996   40   4960   9.9
101.6   0   95   96   1012   29   6220   12.1
47.1   1   46   41   968   19   4570   8.8
48.9   0   106   97   989   40   5930   10.4
85.2   0   90   91   1049   3   5880   12.1

Answer 1

a.

$Robbery_rate = \beta_0 + \beta_1 Southern +\varepsilon\\ \beta_0 = 93.0839\\ \beta_1 = -7.7776 \thinspace p-value = 0.5209\\ R^2 = 0.0092\\$

b.
$Robbery_rate = \beta_0 + \beta_1Southern +\beta_2 Males Per Females+\varepsilon \\ \beta_0 = -183.7163\\ \beta_1 = -2.3617 \thinspace p-value = 0.8515\\ \beta_2 = 0.2797 \thinspace p-value = 0.1770\\ R^2 = 0.0499\\$

c.
$Robbery_rate = \beta_0 + \beta_1Southern +\beta_2 Expend1960+\varepsilon \\ \beta_0 = 0.8946\\ \beta_1 = -15.1736 \thinspace p-value = 0.1077\\ \beta_2 = 0.29927\thinspace p-value = 0.0001\\ R^2 = 0.507\\$

d.

$Robbery_rate = \beta_0 + \beta_1Southern +\beta_2 Expend1960+\beta_3 Males Per Females + \varepsilon \\ \beta_0 = -374.4521\\ \beta_1 = 23.2878 \thinspace p-value = 0.0144\\ \beta_2 = 1.0288\thinspace p-value = 0.0001\\ \beta_2 = 0.3759\thinspace p-value = 0.0092\\ R^2 = 0.5797\\$