The Research question:
Does parental income determines admission and attendance chances into
highly selective U.S. colleges? Controlling for SAT/ACT scores, this
analysis will examine that question at the application, admission and
attendance phases of colleges enrollment.The analysis will also explore
the specific probability of a student getting into a highly selective
U.S. colleges given a parental income range between 70-80.
The data
The data sets used in this project are from a 2023 Harvard
university study conducted by Raj Chetty, David J. Deming, and John
N. Friedman.
The data were collected from 1996 to 2021 and combine anonymized
admissions data from several private and public colleges linked to
parents’ and students’ income tax records and students’ SAT/ACT scores.
The statistics reported in the original study are from 139 colleges in
the U.S. comprised of the following groups: Ivy- Plus colleges(the eight
Ivy leagues colleges + Stanford, Duke, MIT and Chicago), as well as
highly selective private colleges and flagship public colleges. The
primary measure of parental income is total household-level pre-tax
income. The initial study also constructed three distinct measures of
attendance, application and attendance conditional on application. The
cases in the primary data set are U.S. selective colleges and
universities and there are a total of 1946 cases or observations in it.
Our analysis uses the following categorical variable as our response
variable: parent Income Percentile; and the following numeric variables
as our explanatory variables: rel_apply, attend, and rel_attend.
Getting Started: Loading Libraries
Importing and preparing the data
Let’s first take a peek at the distribution of test scores
conditional of parent income.
The table below reveals that there are very few students from low-
income families with sufficient high - test scores to get into highly
selective colleges in the first place.
test_scores <- read_csv("https://raw.githubusercontent.com/Heleinef/Data-Science-Master_Heleine/main/Distribution%20of%20Test%20Scores%20Conditional%20on%20Parent%20Income.csv")
spec(test_scores)
## cols(
## `Parent Income Percenticle:` = col_character(),
## `0-10` = col_character(),
## `0-20` = col_character(),
## `20-30` = col_character(),
## `30-40` = col_character(),
## `40-50` = col_character(),
## `50-60` = col_character(),
## `60-70` = col_character(),
## `70-80` = col_character(),
## `80-90` = col_character(),
## `90-95` = col_character(),
## `95-96` = col_character(),
## `96-97` = col_character(),
## `97-98` = col_character(),
## `98-99` = col_character(),
## `99-99.9` = col_character(),
## `Top 0.1%` = col_character(),
## `Total%` = col_character(),
## ...19 = col_logical()
## )
tibble(test_scores)
The above table shows that among students scoring high on the
SAT/ACT, only 3.7% come from the lowest quintile of parent income.This
means that there are very few students from low- income families with
sufficient high - test scores to get into highly selective colleges to
begin with.
Now, let’s focus on colleges admissions.
names(CollegesAdmissions)
## [1] "super_opeid" "name"
## [3] "par_income_bin" "par_income_lab"
## [5] "attend" "stderr_attend"
## [7] "attend_level" "attend_sat"
## [9] "stderr_attend_sat" "attend_level_sat"
## [11] "rel_apply" "stderr_rel_apply"
## [13] "rel_attend" "stderr_rel_attend"
## [15] "rel_att_cond_app" "rel_apply_sat"
## [17] "stderr_rel_apply_sat" "rel_attend_sat"
## [19] "stderr_rel_attend_sat" "rel_att_cond_app_sat"
## [21] "attend_instate" "stderr_attend_instate"
## [23] "attend_level_instate" "attend_instate_sat"
## [25] "stderr_attend_instate_sat" "attend_level_instate_sat"
## [27] "attend_oostate" "stderr_attend_oostate"
## [29] "attend_level_oostate" "attend_oostate_sat"
## [31] "stderr_attend_oostate_sat" "attend_level_oostate_sat"
## [33] "rel_apply_instate" "stderr_rel_apply_instate"
## [35] "rel_attend_instate" "stderr_rel_attend_instate"
## [37] "rel_att_cond_app_instate" "rel_apply_oostate"
## [39] "stderr_rel_apply_oostate" "rel_attend_oostate"
## [41] "stderr_rel_attend_oostate" "rel_att_cond_app_oostate"
## [43] "rel_apply_instate_sat" "stderr_rel_apply_instate_sat"
## [45] "rel_attend_instate_sat" "stderr_rel_attend_instate_sat"
## [47] "rel_att_cond_app_instate_sat" "rel_apply_oostate_sat"
## [49] "stderr_rel_apply_oostate_sat" "rel_attend_oostate_sat"
## [51] "stderr_rel_attend_oostate_sat" "rel_att_cond_app_oostate_sat"
## [53] "attend_unwgt" "stderr_attend_unwgt"
## [55] "attend_unwgt_level" "attend_unwgt_instate"
## [57] "stderr_attend_unwgt_instate" "attend_unwgt_oostate"
## [59] "stderr_attend_unwgt_oostate" "attend_unwgt_level_instate"
## [61] "attend_unwgt_level_oostate" "rel_attend_unwgt"
## [63] "rel_apply_unwgt" "stderr_rel_attend_unwgt"
## [65] "stderr_rel_apply_unwgt" "rel_att_cond_app_unwgt"
## [67] "rel_attend_unwgt_instate" "rel_attend_unwgt_oostate"
## [69] "stderr_rel_attend_unwgt_instate" "stderr_rel_attend_unwgt_oostate"
## [71] "rel_apply_unwgt_instate" "rel_apply_unwgt_oostate"
## [73] "stderr_rel_apply_unwgt_instate" "stderr_rel_apply_unwgt_oostate"
## [75] "rel_att_cond_app_unwgt_instate" "rel_att_cond_app_unwgt_oostate"
## [77] "public" "flagship"
## [79] "tier" "tier_name"
## [81] "test_band_tier"
# skim main-df
skim(CollegesAdmissions)
Data summary
| Name |
CollegesAdmissions |
| Number of rows |
1946 |
| Number of columns |
81 |
| _______________________ |
|
| Column type frequency: |
|
| character |
6 |
| numeric |
75 |
| ________________________ |
|
| Group variables |
None |
Variable type: character
| name |
0 |
1 |
12 |
49 |
0 |
139 |
0 |
| par_income_lab |
0 |
1 |
4 |
7 |
0 |
14 |
0 |
| public |
0 |
1 |
6 |
7 |
0 |
2 |
0 |
| tier |
0 |
1 |
8 |
40 |
0 |
6 |
0 |
| tier_name |
0 |
1 |
8 |
40 |
0 |
6 |
0 |
| test_band_tier |
0 |
1 |
6 |
21 |
0 |
6 |
0 |
Variable type: numeric
| super_opeid |
0 |
1.00 |
2528.51 |
1344.92 |
108.00 |
1536.00 |
2536.00 |
3223.00 |
11649.00 |
▇▇▁▁▁ |
| par_income_bin |
0 |
1.00 |
78.17 |
28.04 |
10.00 |
65.00 |
94.00 |
98.50 |
100.00 |
▁▁▁▂▇ |
| attend |
2 |
1.00 |
0.00 |
0.01 |
0.00 |
0.00 |
0.00 |
0.01 |
0.05 |
▇▁▁▁▁ |
| stderr_attend |
0 |
1.00 |
0.00 |
0.00 |
0.00 |
0.00 |
0.00 |
0.00 |
0.01 |
▇▁▁▁▁ |
| attend_level |
0 |
1.00 |
0.00 |
0.00 |
0.00 |
0.00 |
0.00 |
0.01 |
0.02 |
▇▂▁▁▁ |
| attend_sat |
294 |
0.85 |
0.00 |
0.00 |
0.00 |
0.00 |
0.00 |
0.01 |
0.04 |
▇▁▁▁▁ |
| stderr_attend_sat |
278 |
0.86 |
0.00 |
0.00 |
0.00 |
0.00 |
0.00 |
0.00 |
0.00 |
▇▁▁▁▁ |
| attend_level_sat |
0 |
1.00 |
0.00 |
0.00 |
0.00 |
0.00 |
0.00 |
0.00 |
0.02 |
▇▂▁▁▁ |
| rel_apply |
0 |
1.00 |
1.17 |
0.56 |
0.07 |
0.83 |
1.05 |
1.34 |
5.87 |
▇▃▁▁▁ |
| stderr_rel_apply |
0 |
1.00 |
0.04 |
0.04 |
0.01 |
0.02 |
0.03 |
0.05 |
0.53 |
▇▁▁▁▁ |
| rel_attend |
2 |
1.00 |
1.27 |
0.93 |
0.01 |
0.76 |
1.03 |
1.45 |
10.26 |
▇▁▁▁▁ |
| stderr_rel_attend |
0 |
1.00 |
0.11 |
0.12 |
0.00 |
0.05 |
0.08 |
0.13 |
1.57 |
▇▁▁▁▁ |
| rel_att_cond_app |
2 |
1.00 |
1.03 |
0.27 |
0.05 |
0.88 |
1.00 |
1.15 |
3.06 |
▁▇▂▁▁ |
| rel_apply_sat |
278 |
0.86 |
1.15 |
0.51 |
0.19 |
0.79 |
1.06 |
1.36 |
4.70 |
▇▆▁▁▁ |
| stderr_rel_apply_sat |
278 |
0.86 |
0.07 |
0.05 |
0.02 |
0.03 |
0.06 |
0.09 |
0.56 |
▇▁▁▁▁ |
| rel_attend_sat |
294 |
0.85 |
1.19 |
0.75 |
0.02 |
0.71 |
0.99 |
1.48 |
8.03 |
▇▂▁▁▁ |
| stderr_rel_attend_sat |
278 |
0.86 |
0.18 |
0.13 |
0.00 |
0.08 |
0.15 |
0.24 |
0.95 |
▇▃▁▁▁ |
| rel_att_cond_app_sat |
294 |
0.85 |
1.01 |
0.35 |
0.04 |
0.82 |
0.98 |
1.17 |
3.85 |
▂▇▁▁▁ |
| attend_instate |
1334 |
0.31 |
0.13 |
0.08 |
0.00 |
0.06 |
0.13 |
0.18 |
0.41 |
▇▇▅▂▁ |
| stderr_attend_instate |
1334 |
0.31 |
0.01 |
0.01 |
0.00 |
0.00 |
0.01 |
0.01 |
0.08 |
▇▁▁▁▁ |
| attend_level_instate |
1232 |
0.37 |
0.12 |
0.07 |
0.02 |
0.05 |
0.13 |
0.17 |
0.36 |
▇▇▆▂▁ |
| attend_instate_sat |
1334 |
0.31 |
0.13 |
0.10 |
0.00 |
0.04 |
0.11 |
0.18 |
0.66 |
▇▃▂▁▁ |
| stderr_attend_instate_sat |
1334 |
0.31 |
0.02 |
0.02 |
0.00 |
0.00 |
0.01 |
0.02 |
0.15 |
▇▁▁▁▁ |
| attend_level_instate_sat |
1232 |
0.37 |
0.12 |
0.08 |
0.01 |
0.03 |
0.11 |
0.16 |
0.39 |
▇▇▃▁▁ |
| attend_oostate |
1336 |
0.31 |
0.00 |
0.00 |
0.00 |
0.00 |
0.00 |
0.00 |
0.02 |
▇▁▁▁▁ |
| stderr_attend_oostate |
1334 |
0.31 |
0.00 |
0.00 |
0.00 |
0.00 |
0.00 |
0.00 |
0.00 |
▇▂▁▁▁ |
| attend_level_oostate |
1232 |
0.37 |
0.00 |
0.00 |
0.00 |
0.00 |
0.00 |
0.00 |
0.00 |
▇▆▂▁▁ |
| attend_oostate_sat |
1346 |
0.31 |
0.00 |
0.00 |
0.00 |
0.00 |
0.00 |
0.00 |
0.01 |
▇▁▁▁▁ |
| stderr_attend_oostate_sat |
1334 |
0.31 |
0.00 |
0.00 |
0.00 |
0.00 |
0.00 |
0.00 |
0.00 |
▇▂▂▁▁ |
| attend_level_oostate_sat |
1232 |
0.37 |
0.00 |
0.00 |
0.00 |
0.00 |
0.00 |
0.00 |
0.00 |
▇▆▃▂▂ |
| rel_apply_instate |
1334 |
0.31 |
1.01 |
0.17 |
0.27 |
0.91 |
1.00 |
1.12 |
1.70 |
▁▁▇▃▁ |
| stderr_rel_apply_instate |
1334 |
0.31 |
0.03 |
0.02 |
0.01 |
0.01 |
0.02 |
0.03 |
0.13 |
▇▂▁▁▁ |
| rel_attend_instate |
1334 |
0.31 |
1.03 |
0.30 |
0.14 |
0.86 |
1.01 |
1.20 |
2.19 |
▁▆▇▂▁ |
| stderr_rel_attend_instate |
1334 |
0.31 |
0.06 |
0.04 |
0.01 |
0.03 |
0.05 |
0.07 |
0.31 |
▇▂▁▁▁ |
| rel_att_cond_app_instate |
1334 |
0.31 |
1.01 |
0.19 |
0.37 |
0.90 |
1.00 |
1.11 |
1.70 |
▁▃▇▂▁ |
| rel_apply_oostate |
1334 |
0.31 |
1.19 |
0.55 |
0.32 |
0.75 |
1.10 |
1.54 |
4.14 |
▇▆▂▁▁ |
| stderr_rel_apply_oostate |
1334 |
0.31 |
0.05 |
0.04 |
0.01 |
0.02 |
0.03 |
0.06 |
0.26 |
▇▂▁▁▁ |
| rel_attend_oostate |
1336 |
0.31 |
1.30 |
0.87 |
0.24 |
0.63 |
1.08 |
1.74 |
5.86 |
▇▃▁▁▁ |
| stderr_rel_attend_oostate |
1334 |
0.31 |
0.14 |
0.11 |
0.00 |
0.06 |
0.11 |
0.18 |
0.75 |
▇▃▁▁▁ |
| rel_att_cond_app_oostate |
1336 |
0.31 |
1.02 |
0.28 |
0.34 |
0.82 |
1.01 |
1.18 |
2.42 |
▂▇▃▁▁ |
| rel_apply_instate_sat |
1334 |
0.31 |
1.02 |
0.20 |
0.31 |
0.90 |
1.00 |
1.16 |
1.57 |
▁▂▇▅▁ |
| stderr_rel_apply_instate_sat |
1334 |
0.31 |
0.06 |
0.04 |
0.02 |
0.03 |
0.05 |
0.07 |
0.28 |
▇▂▁▁▁ |
| rel_attend_instate_sat |
1334 |
0.31 |
1.07 |
0.44 |
0.09 |
0.79 |
1.00 |
1.27 |
2.95 |
▂▇▃▁▁ |
| stderr_rel_attend_instate_sat |
1334 |
0.31 |
0.14 |
0.10 |
0.03 |
0.07 |
0.11 |
0.16 |
0.71 |
▇▂▁▁▁ |
| rel_att_cond_app_instate_sat |
1334 |
0.31 |
1.03 |
0.30 |
0.15 |
0.86 |
1.00 |
1.15 |
2.17 |
▁▆▇▁▁ |
| rel_apply_oostate_sat |
1334 |
0.31 |
1.30 |
0.70 |
0.28 |
0.73 |
1.12 |
1.73 |
4.21 |
▇▅▃▁▁ |
| stderr_rel_apply_oostate_sat |
1334 |
0.31 |
0.11 |
0.09 |
0.02 |
0.04 |
0.08 |
0.15 |
0.67 |
▇▂▁▁▁ |
| rel_attend_oostate_sat |
1346 |
0.31 |
1.49 |
1.21 |
0.06 |
0.55 |
1.07 |
2.09 |
9.01 |
▇▃▁▁▁ |
| stderr_rel_attend_oostate_sat |
1334 |
0.31 |
0.34 |
0.31 |
0.00 |
0.11 |
0.24 |
0.47 |
2.14 |
▇▃▁▁▁ |
| rel_att_cond_app_oostate_sat |
1346 |
0.31 |
1.05 |
0.42 |
0.10 |
0.76 |
0.99 |
1.25 |
2.59 |
▂▇▅▁▁ |
| attend_unwgt |
1 |
1.00 |
0.00 |
0.00 |
0.00 |
0.00 |
0.00 |
0.00 |
0.02 |
▇▁▁▁▁ |
| stderr_attend_unwgt |
0 |
1.00 |
0.00 |
0.00 |
0.00 |
0.00 |
0.00 |
0.00 |
0.00 |
▇▁▁▁▁ |
| attend_unwgt_level |
0 |
1.00 |
0.00 |
0.00 |
0.00 |
0.00 |
0.00 |
0.00 |
0.00 |
▇▂▂▁▁ |
| attend_unwgt_instate |
1334 |
0.31 |
0.09 |
0.08 |
0.00 |
0.03 |
0.07 |
0.14 |
0.35 |
▇▃▂▁▁ |
| stderr_attend_unwgt_instate |
1334 |
0.31 |
0.01 |
0.01 |
0.00 |
0.00 |
0.00 |
0.01 |
0.04 |
▇▁▁▁▁ |
| attend_unwgt_oostate |
1337 |
0.31 |
0.00 |
0.00 |
0.00 |
0.00 |
0.00 |
0.00 |
0.01 |
▇▁▁▁▁ |
| stderr_attend_unwgt_oostate |
1334 |
0.31 |
0.00 |
0.00 |
0.00 |
0.00 |
0.00 |
0.00 |
0.00 |
▇▂▁▁▁ |
| attend_unwgt_level_instate |
1232 |
0.37 |
0.07 |
0.05 |
0.01 |
0.02 |
0.07 |
0.10 |
0.24 |
▇▆▂▂▁ |
| attend_unwgt_level_oostate |
1232 |
0.37 |
0.00 |
0.00 |
0.00 |
0.00 |
0.00 |
0.00 |
0.00 |
▇▅▃▁▁ |
| rel_attend_unwgt |
1 |
1.00 |
2.48 |
3.10 |
0.00 |
0.62 |
1.41 |
2.94 |
30.80 |
▇▁▁▁▁ |
| rel_apply_unwgt |
0 |
1.00 |
1.98 |
1.77 |
0.09 |
0.71 |
1.42 |
2.62 |
13.77 |
▇▂▁▁▁ |
| stderr_rel_attend_unwgt |
0 |
1.00 |
0.20 |
0.35 |
0.00 |
0.03 |
0.07 |
0.22 |
3.21 |
▇▁▁▁▁ |
| stderr_rel_apply_unwgt |
0 |
1.00 |
0.06 |
0.09 |
0.00 |
0.01 |
0.03 |
0.07 |
1.06 |
▇▁▁▁▁ |
| rel_att_cond_app_unwgt |
1 |
1.00 |
1.06 |
0.39 |
0.01 |
0.84 |
1.00 |
1.20 |
4.95 |
▇▇▁▁▁ |
| rel_attend_unwgt_instate |
1334 |
0.31 |
1.30 |
0.73 |
0.18 |
0.79 |
1.10 |
1.64 |
4.70 |
▇▆▂▁▁ |
| rel_attend_unwgt_oostate |
1337 |
0.31 |
1.82 |
1.67 |
0.09 |
0.53 |
1.29 |
2.82 |
12.10 |
▇▃▁▁▁ |
| stderr_rel_attend_unwgt_instate |
1334 |
0.31 |
0.07 |
0.05 |
0.01 |
0.03 |
0.05 |
0.09 |
0.34 |
▇▂▁▁▁ |
| stderr_rel_attend_unwgt_oostate |
1334 |
0.31 |
0.16 |
0.15 |
0.00 |
0.04 |
0.10 |
0.24 |
0.93 |
▇▃▁▁▁ |
| rel_apply_unwgt_instate |
1334 |
0.31 |
1.19 |
0.43 |
0.35 |
0.86 |
1.10 |
1.49 |
2.40 |
▃▇▅▃▁ |
| rel_apply_unwgt_oostate |
1334 |
0.31 |
1.57 |
1.08 |
0.17 |
0.69 |
1.31 |
2.26 |
7.07 |
▇▅▁▁▁ |
| stderr_rel_apply_unwgt_instate |
1334 |
0.31 |
0.03 |
0.02 |
0.00 |
0.01 |
0.02 |
0.04 |
0.12 |
▇▃▁▁▁ |
| stderr_rel_apply_unwgt_oostate |
1334 |
0.31 |
0.05 |
0.04 |
0.00 |
0.01 |
0.03 |
0.07 |
0.31 |
▇▃▁▁▁ |
| rel_att_cond_app_unwgt_instate |
1334 |
0.31 |
1.04 |
0.27 |
0.40 |
0.86 |
1.01 |
1.17 |
2.09 |
▂▇▅▁▁ |
| rel_att_cond_app_unwgt_oostate |
1337 |
0.31 |
1.02 |
0.33 |
0.19 |
0.78 |
1.00 |
1.21 |
2.14 |
▂▇▇▂▁ |
| flagship |
0 |
1.00 |
0.21 |
0.41 |
0.00 |
0.00 |
0.00 |
0.00 |
1.00 |
▇▁▁▁▂ |
Summary statistics
# Summary - Colleges Admissions
admissions_summary <- CollegesAdmissions %>%
group_by(par_income_lab) %>%
summarise(across(.cols=contains("att"), list(mean, standard_deviation = sd)))
admissions_summary
Defining elite U.S. elites colleges
elite_colleges <- CollegesAdmissions %>%
group_by(tier_name) %>%
summarise
elite_colleges
Relative application rates at Highly selective U.S. colleges by
parental income
# Relative application rates at Highly selective U.S. colleges by parental income
ggplot(data = CollegesAdmissions, aes(x = par_income_lab, y = rel_apply, color = 3))+
geom_boxplot()+
labs(
x = "Parental Income Percentile",
y = "Relative application Rates",
title = "Relative application Rates at Highly Selective U.S. Colleges by Parental Income",
fill = "tier_name"
)
###The boxplot above shows that, Discrepancies in admissions to elite
U.S.colleges based on parental income are noticeable right from the
start, at the application phase.Controling for SAT/ACT scores, the
boxplot reveals that students from high parental income percentile apply
to elite colleges 5 times more than do their counterparts from the low
and middle parental income percentile.
ggplot(data = CollegesAdmissions, aes(x = par_income_lab, y = rel_apply, color = 3))+
geom_boxplot()+
labs(
x = "Parental Income Percentile",
y = "Relative application Rates",
title = "Relative application Rates at Highly Selective U.S. Colleges by Parental Income",
fill = "tier_name"
)+ facet_wrap(~tier_name)
Attendance rates at Highly selective colleges by parental
income
# Attendance rates at Highly selective colleges by parental income
ggplot(data = CollegesAdmissions, aes(x = par_income_lab, y= attend, color = 3))+
geom_boxplot()+
labs(
x = "Parental Income Percentile",
y = "Attendance Rate",
title = "Attendance Rates at Highly Selective U.S. Colleges by Parental Income",
fill = "tier_name"
)
## Warning: Removed 2 rows containing non-finite values (`stat_boxplot()`).
The above boxplot reveals that most students from higher parental
Income percentile attend highly selective U.S.colleges followed by
students from very low parental income percentile. Students from middle
income family count for 2% of attendance and elite U.S. colleges.
# Attendance rates at Highly selective colleges by parental income
ggplot(data = CollegesAdmissions, aes(x = par_income_lab, y= attend, color = 3))+
geom_boxplot()+
labs(
x = "Parental Income Percentile",
y = "Attendance Rate",
title = "Attendance Rates at Highly Selective U.S. Colleges by Parental Income",
fill = "tier_name"
)+ facet_wrap(~tier_name)
## Warning: Removed 2 rows containing non-finite values (`stat_boxplot()`).
Relative attendance rates at Highly selective colleges by parental
income.
# Relative attendance rates at Highly selective colleges by parental income
ggplot(data = CollegesAdmissions, aes(x = par_income_lab, y= rel_attend, color = 3))+
geom_boxplot()+
labs(
x = "Parental Income Percentile",
y = "Relative Attendance Rate",
title = "Relative Attendance Rates at Highly Selective U.S. Colleges by Parental Income",
fill = "tier_name"
)
## Warning: Removed 2 rows containing non-finite values (`stat_boxplot()`).
Students from middle parental income ranges(40-60, 70-80, 90-95) have
less than 2% chances to attend highly selective U.S. colleges, while
students from top earning families have more than 5 times to get
admitted than their counterparts from the middle and lower income
families.
ggplot(CollegesAdmissions, aes(x = par_income_lab, y= rel_attend, color = tier_name, fill = tier_name))+
geom_boxplot()+
labs(
x = "Parental Income Percentile",
y = "Relative Attendance Rate",
title = "Relative Attendance Rates at Highly Selective U.S.Colleges by Parental Income",
fill = "tier_name"
)+ facet_wrap(~tier_name)
## Warning: Removed 2 rows containing non-finite values (`stat_boxplot()`).
Results of relative attendance when filtering for only complete
cases
# let's calculate relative rates by parental income based on the filtered data
ggplot(data = CollegesAdmissions_new, aes (x= par_income_lab, y = rel_attend, fill = tier_name, color = 3))+ geom_boxplot() +
labs(
x = "Parental Income Percentile",
y = "Relative Attendance Rate",
title = "relative Attendance Rates by Parental Income and tier_name"
) + facet_wrap(~tier_name)
Correlation tests:
Let’s explore the strengths of the correlations existing between our
response variable(parental income percentile) and two other variables
from the Colleges Admission data set in order to understand which factor
most explains difference in attendance level at selective colleges.
Positive but weak correlation between parental income and relative
application to elite colleges: 0.388087
# Relative application to elite colleges and parental income
cor.test(CollegesAdmissions$par_income_bin,CollegesAdmissions$rel_apply)
##
## Pearson's product-moment correlation
##
## data: CollegesAdmissions$par_income_bin and CollegesAdmissions$rel_apply
## t = 18.566, df = 1944, p-value < 2.2e-16
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
## 0.3496821 0.4251898
## sample estimates:
## cor
## 0.388087
Linear regression
## linear modeling
CollegesAdmissions_lm1 <- lm(par_income_bin ~ rel_apply, data = CollegesAdmissions)
summary(CollegesAdmissions_lm1)
##
## Call:
## lm(formula = par_income_bin ~ rel_apply, data = CollegesAdmissions)
##
## Residuals:
## Min 1Q Median 3Q Max
## -83.55 -8.74 10.26 17.74 43.20
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 55.432 1.358 40.83 <2e-16 ***
## rel_apply 19.392 1.044 18.57 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 25.85 on 1944 degrees of freedom
## Multiple R-squared: 0.1506, Adjusted R-squared: 0.1502
## F-statistic: 344.7 on 1 and 1944 DF, p-value: < 2.2e-16
There is a positive but moderate relationship between parental
income and rel_att_cond_app_unwgt
cor.test(CollegesAdmissions$par_income_bin,CollegesAdmissions$rel_att_cond_app_unwgt)
##
## Pearson's product-moment correlation
##
## data: CollegesAdmissions$par_income_bin and CollegesAdmissions$rel_att_cond_app_unwgt
## t = 21.174, df = 1943, p-value < 2.2e-16
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
## 0.3961666 0.4684207
## sample estimates:
## cor
## 0.4329889
Linear regression
# linear modeling
CollegesAdmissions_lm2 <- lm(par_income_bin~rel_att_cond_app_unwgt, data = CollegesAdmissions)
summary(CollegesAdmissions_lm2)
##
## Call:
## lm(formula = par_income_bin ~ rel_att_cond_app_unwgt, data = CollegesAdmissions)
##
## Residuals:
## Min 1Q Median 3Q Max
## -100.645 -12.019 8.387 17.607 55.015
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 44.811 1.676 26.73 <2e-16 ***
## rel_att_cond_app_unwgt 31.494 1.487 21.17 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 25.28 on 1943 degrees of freedom
## (1 observation deleted due to missingness)
## Multiple R-squared: 0.1875, Adjusted R-squared: 0.1871
## F-statistic: 448.3 on 1 and 1943 DF, p-value: < 2.2e-16
Evaluating the predictive value of the linear models
# Predicted values (lm1)
CollegesAdmissions_lm1_predicted <- predict(CollegesAdmissions_lm1)
glimpse(CollegesAdmissions_lm1_predicted)
## Named num [1:1946] 68.4 68.6 69.1 69.3 69.4 ...
## - attr(*, "names")= chr [1:1946] "1" "2" "3" "4" ...
# Predicted values (lm2)
CollegesAdmissions_lm2_predicted <- predict(CollegesAdmissions_lm2)
glimpse(CollegesAdmissions_lm2_predicted)
## Named num [1:1945] 81.4 75.8 82.1 85.2 85.4 ...
## - attr(*, "names")= chr [1:1945] "1" "2" "3" "4" ...
Evaluating the probability of getting accepted into a U.S.elite
college with a middle class parental income range between 70-80
Conditional Probability: non - academic requirements at elites
significantly reduces the probability that a student from a middle
income range will get into an elite U.S. college
Given that elite U.S. colleges have different admission criteria
beyond academic performance, such as extracurricular activities and/or
legacy preferences. The actual probability of admission may vary
depending on these criteria.Therefore the probability of admission for a
student from a middle class parental income between 70-80 being admitted
in an elite college could be read as followed:
P(A | B) = P(A and B) / P(B)
where A represents the event of an applicant being admitted, and B
represents the event of an applicant meeting the admission criteria.
Therefore, the probability of an applicant being admitted given that
they meet the admission criteria is given by:
P(admitted | meets criteria) = P(admitted and meets criteria) /
P(meets criteria)
where P(admitted | meets criteria) is the probability of an applicant
being admitted given that they meet the admission criteria, P(admitted
and meets criteria) is the probability of an applicant being admitted
and meeting the admission criteria, and P(meets criteria) is the
probability of an applicant meeting the admission criteria.
Main Findings:
1.We found disparities in application, admission and attendance rates
at american elite colleges
2.Among students scoring high on the SAT/ACT, only 3.7% come from the
lowest quintile of parent income.
3.From the factors we measured, we found a positive but moderate
relationship between parental income and
rel_att_cond_app_unwgt(0.4329889). We also found a positive but weak
correlation between parental income and relative application to elite
colleges: 0.388087
4.Students from middle parental income ranges(40-60, 70-80, 90-95)
have less than 2% chances to attend highly selective U.S. colleges
5.There are very few students from low - income families with
sufficient high - test scores to get into highly selective colleges.
6.There is a substantial under matching of middle – income & low-
income students at Ivy-Plus schools.
7.Ivy-Plus colleges are more than twice as likely to admit a student
from a high – income family as compared to low or middle – income
families with comparable SAT/ACT scores.
8.Controlling for SAT/ACT scores, the probability of attending an
elite private college is 77 times higher for children in the top 1%
compared to students fromthe bottom 20%