Problem on the Interpretation of Specfic Regression Equation Variables

This is a problem on interpretation of regression equations which have categorical explanatory variables where slopes on non-categorical variables do not depend on the category. This model assumes that hyperplanes are parallel for different categories, and the regression coefficients for the binary dummy variables can be used to determine distances between hyperplanes for different categories.

Context of data set: American post-secondary schools in 2014-2015, where the annual in-state tuition is less than \$ 20000. The response variable is the total number of applicants (in thousands). There were many explanatory variables in the complete data set, but only a few are included here. Region is a categorical variable for different parts of the United States. There is some merging of categories compared with the original data set. For your multiple regression output, check that your estimates are interpretable before you submit answers.

For your subset of the university applicant data set, the response variable is: applicants, in thousands**(for the questions below, convert the response with natural logarithm).**

applicants=c(20.756, 8.754, 33.736, 5.111, 20.677, 18.107, 10.04, 36.101, 50.299, 24.988, 38.785, 60.543, 36.362, 14.223, 25.438, 28.518, 32.19, 13.799, 9.679, 3.542, 12.92, 13.758, 5.017, 20.934, 16.958, 10.039, 20.923, 86.537, 44.76, 1.751, 33.211, 10.332, 31.28, 5.465, 19.814, 28.662, 5.002, 18.42, 39.896, 73.448, 35.822, 7.075, 26.496, 16.125, 8.832, 20.443, 10.991, 10.245, 24.233, 7.408, 49.776, 11.552, 25.194, 14.944, 31.332, 14.116, 40.727, 66.515, 20.175, 14.887, 14.552, 10.217, 14.933, 33.714, 4.777, 10.394, 5.713, 10.111, 11.258, 20.918, 33.226, 20.744, 21.359, 7.532, 5.345)

The explanatory variables are: (i) per.admit (percentage admitted) per.admit=c(52, 95, 51, 76, 72, 60, 61, 58, 50, 53, 40, 40, 76, 77, 57, 44, 71, 84, 63, 89, 70, 75, 49, 74, 83, 42, 77, 19, 45, 59, 66, 63, 50, 83, 51, 46, 84, 80, 59, 33, 59, 57, 62, 84, 84, 52, 81, 61, 73, 87, 32, 83, 68, 66, 28, 78, 56, 37, 53, 82, 33, 60, 76, 41, 63, 95, 83, 52, 83, 56, 50, 73, 75, 79, 93)

(ii) num.enroll (enrollment, in thousands) num.enroll=c(21.857, 14.534, 36.047, 7.099, 29.203, 20.655, 18.647, 21.498, 47.04, 41.938, 51.313, 34.508, 46.416, 27.511, 42.598, 16.695, 61.642, 14.982, 20.517, 10.061, 30.69, 30.848, 10.241, 29.217, 25.912, 12.602, 16.571, 41.845, 51.147, 11.645, 50.081, 11.314, 26.541, 15.805, 20.611, 49.459, 16.936, 15.117, 39.752, 30.709, 45.14, 10.725, 37.485, 29.477, 19.934, 33.989, 31.515, 39.74, 12.856, 28.515, 43.625, 11.286, 22.68, 30.297, 29.135, 28.886, 17.866, 30.051, 49.61, 28.686, 8.437, 13.979, 28.628, 24.607, 10.646, 20.626, 14.747, 9.233, 13.183, 35.197, 60.767, 31.224, 24.096, 19.507, 13.952)

(iii) stfacratio (student/faculty ratio) stfacratio=c(16, 19, 21, 13, 18, 15, 17, 19, 17, 25, 17, 18, 17, 18, 18, 20, 20, 19, 21, 17, 19, 17, 15, 18, 17, 23, 17, 16, 17, 13, 17, 11, 16, 20, 15, 21, 17, 19, 12, 19, 19, 18, 14, 21, 22, 16, 17, 26, 15, 22, 12, 16, 15, 24, 13, 19, 18, 19, 26, 15, 12, 19, 23, 16, 17, 20, 19, 18, 20, 18, 31, 16, 19, 13, 18)

(iv) avg.grant (average grant for financial aid, in thousands) avg.grant=c(8.684, 5.969, 10.736, 10.316, 11.848, 9.097, 7.937, 16.227, 6.541, 5.976, 8.727, 15.528, 8.834, 7.609, 8.036, 6.838, 8.731, 4.582, 5.421, 4.412, 7.959, 7.736, 6.974, 5.715, 8.901, 5.678, 9.507, 17.423, 7.511, 6.601, 9.747, 9.924, 10.265, 5.826, 8.076, 6.05, 5.896, 11.579, 9.322, 16.638, 11.818, 7.745, 7.327, 6.229, 5.462, 9.322, 6.726, 6.234, 12.615, 4.965, 14.671, 7.719, 8.788, 5.343, 13.447, 7.035, 17.287, 17.09, 5.055, 10.526, 14.779, 7.879, 7.118, 7.621, 11.767, 7.338, 4.89, 7.215, 7.746, 8.227, 4.616, 7.95, 6.451, 7.576, 5.036)

(v) grad.rate (graduation rate, maybe this means within 4 or 5 years) grad.rate=c(82, 50, 67, 66, 58, 57, 54, 66, 86, 57, 79, 81, 75, 58, 82, 79, 80, 49, 50, 45, 38, 56, 40, 63, 68, 45, 63, 92, 73, 19, 79, 69, 82, 37, 54, 85, 34, 78, 70, 86, 84, 43, 66, 52, 54, 71, 59, 40, 76, 41, 91, 59, 80, 41, 89, 61, 74, 86, 49, 67, 93, 61, 28, 70, 54, 46, 53, 12, 43, 79, 65, 83, 67, 56, 49)

(vi) region (5 categories are FarWest, Gl.NE for GreatLakesNewEngland, Mid for Middle/Central longitude, Southeast, West) region=c('Southeast', 'Mid', 'Southeast', 'GLNE', 'Southeast', 'GLNE', 'Southeast', 'FarWest', 'Mid', 'Southeast', 'West', 'FarWest', 'GLNE', 'Southeast', 'GLNE', 'Mid', 'West', 'West', 'Southeast', 'Mid', 'GLNE', 'Southeast', 'Southeast', 'GLNE', 'Southeast', 'Southeast', 'GLNE', 'FarWest', 'Mid', 'Southeast', 'GLNE', 'Mid', 'GLNE', 'Southeast', 'GLNE', 'Southeast', 'GLNE', 'GLNE', 'GLNE', 'FarWest', 'GLNE', 'Southeast', 'Mid', 'GLNE', 'FarWest', 'Southeast', 'West', 'West', 'GLNE', 'FarWest', 'GLNE', 'GLNE', 'Mid', 'Southeast', 'Southeast', 'FarWest', 'FarWest', 'FarWest', 'Southeast', 'FarWest', 'Southeast', 'Mid', 'West', 'Mid', 'Mid', 'GLNE', 'Mid', 'West', 'GLNE', 'Southeast', 'Southeast', 'Southeast', 'FarWest', 'FarWest', 'West')

You are to fit a multiple regression model with the response variable log(applicants), the natural logarithm of "applicants" and the 6 explanatory variables given above. After you have copied the above R vectors into your R session, you can get a dataframe with:

(Please use 3 decimal places for the answers below which are not integer-valued. )

For the regression being requested, you should find the most or all of the coefficients for per.admit, num.enroll, stfacratio, avg.grant, grad.rate to be statistically significant. Some of the regions might be significantly different from others but not all pairs of regions are significantly different from each other.

To answer the parts (a) and (b) below, two separate regressions could be done (with 2 different regions as the baseline categories). If you want to challenge yourself to answer them both based on one application of lm(), you need to use the cov.unscaled component of the summary of an lm object.

Part a) What is the estimate of the signed distance of the hyperplane for region Southeast relative to Farwest and the SE?

Part b) What is the estimate of the signed distance of the hyperplane for region Southeast relative to GLNE and the SE?

Part c) What is the adjusted R^2?

Part d) What is the residual SD (residual SE in r)?

SO FAR, I have attempted to solve this problem through RStudio. I have tried:

lm(formula = applicants ~ per.admit + num.enroll + stfacratio + avg.grant + grad.rate + region) summary(lm(formula = applicants ~ per.admit + num.enroll + stfacratio + avg.grant + grad.rate + region))

This is the output from Rstudio:

Call: lm(formula = applicants ~ per.admit + num.enroll + stfacratio + avg.grant + grad.rate + region)

Residuals: Min 1Q Median 3Q Max -16.7951 -4.1694 0.5629 4.0614 18.1934

Coefficients: Estimate Std. Error t value Pr(>|t|)
(Intercept) 4.35610 11.13133 0.391 0.696828
per.admit -0.31548 0.06663 -4.735 1.23e-05 * num.enroll 0.46485 0.08842 5.257 1.74e-06 stfacratio 0.45562 0.30644 1.487 0.141899
avg.grant 1.74970 0.43346 4.037 0.000145
grad.rate 0.13864 0.07607 1.823 0.072957 .
regionGLNE -5.20801 2.83067 -1.840 0.070356 .
regionMid -3.67848 3.49285 -1.053 0.296175
regionSoutheast -12.56132 2.97406 -4.224 7.64e-05 *

regionWest -7.63882 3.65328 -2.091 0.040447 *

Signif. codes: 0 ‘’ 0.001 ‘’ 0.01 ‘’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 7.105 on 65 degrees of freedom Multiple R-squared: 0.8415, Adjusted R-squared: 0.8196 F-statistic: 38.35 on 9 and 65 DF, p-value: < 2.2e-16

As of now, none of the values that I obtained are correct.