Question 2 [50 points] MATH 208 Final Exam Solution

We will re-use the same midwest_modified data that was used in Question 1, with all the modifications from the other question parts. The description is repeated below for your convenience.

str(midwest_modified)


spec_tbl_df [437 x 11] (S3: spec_tbl_df/tbl_df/tbl/data.frame)

$ county
: chr [1:437]
"ADAMS" "ALEXANDER" "BOND" "BOONE" ...
$ state
: chr [1:437]
"IL" "IL" "IL" "IL" ...
$ popdensity
: num [1:437]
1271 759 681 1812 324 ...
$ popwhite
: num [1:437]
63917 7054 14477 29344 5264 ...
$ popblack
: num [1:437]
1702 3496 429 127 547 ...
$ popamerindian: num [1:437]
98 19 35 46 14 65 8 30 8 331 ...
$ popasian
: num [1:437]
249
48 16 150 5 ...
$ popother
: num [1:437]
124
9 34 1139 6 ...
$ inmetro
: num [1:437]
0 0
01000001...
$ Metro
: chr [1:437]
"NonMetro" "NonMetro" "NonMetro" "Metro" ...
$ HighDens
: chr [1:437]
"NotHigh" "NotHigh" "NotHigh" "High" ...
    • attr(*, "spec")=

.. cols(

..   county = col_character(),

..   state = col_character(),

..   popdensity = col_double(),

..   popwhite = col_double(),

..   popblack = col_double(),

..   popamerindian = col_double(),

..   popasian = col_double(),

..   popother = col_double(),

..   inmetro = col_double(),

..   Metro = col_character(),

..   HighDens = col_character()

.. )

midwest_modified %>% slice(1:5) %>%


select(county:popblack)

# A tibble:
5 x 5







county
state popdensity popwhite popblack




<chr>
<chr>
<dbl>
<dbl>
<dbl>



1
ADAMS
IL
1271.
63917
1702



2
ALEXANDER
IL
759
7054
3496



3
BOND
IL
681.
14477
429



4
BOONE
IL
1812.
29344
127



5
BROWN
IL
324.
5264
547








midwest_modified %>% slice(1:5) %>%





select(county,popamerindian:HighDens)



# A tibble:
5 x 7







county
popamerindian popasian popother inmetro Metro
HighDens

<chr>

<dbl>
<dbl>
<dbl>
<dbl> <chr>
<chr>
1
ADAMS

98
249
124
0
NonMetro
NotHigh
2
ALEXANDER

19
48
9
0
NonMetro
NotHigh
3
BOND

35
16
34
0
NonMetro
NotHigh
4
BOONE

46
150
1139
1
Metro
High
5
BROWN

14
5
6
0
NonMetro
NotHigh

The dataset contains population data from midwest counties in five states in the United States from an unspecified year. There are identifying variables for both the county (the name) and the state (the postal abbreviation).

1
MATH 208 Final Exam    December 18th – 21st,  



The variable popdensity is a measure of density (population per unspecified area units). The variable inmetro is equal to 1 if the county is classified as a metropolitan area and 0 otherwise. The other variables contain counts of population size within self-identified racial classifications.




CONTINUED ON NEXT PAGE




























































2
MATH 208 Final Exam    December 18th – 21st,  



    (a) [6 pts] Below are partially obscured code and three plots of the values of the log (base 10) of the population density for all counties:

p1<-ggplot(midwest_modified,aes(x=popdensity)) + geom_XXXXX(nbins=30,fill="white",col="black") + ggtitle("Plot 1") + theme_bw() + scale_x_log10()

p2<-ggplot(midwest_modified,aes(x=popdensity)) + geom_YYYYY() + ggtitle("Plot 2") + theme_bw()+ scale_x_log10()

p3<-ggplot(midwest_modified,aes(x=popdensity)) + geom_ZZZZZ() + ggtitle("Plot 3") + theme_bw()+ scale_x_log10()

grid.arrange(grobs=list(p1,p2,p3),nrow=3,ncol=1)

CONTINUED ON NEXT PAGE

Plot 1


60



count
40




20




0




1e+02
1e+03
1e+04
1e+05



popdensity



Plot 2


density

0.75

0.50

0.25

0.00

1e+02    1e+03    1e+04    1e+05

popdensity

Plot 3

0.4


0.2

0.0

−0.2

−0.4

































1e+02

1e+03

1e+04
1e+05









popdensity



Identify these three plots by name:







Plot 1
Plot 2

Plot 3




    (b) [10 pts] Now we make the same plots, but for each state. Do you believe there is evidence of an association between state and population density? In particular, do we see differences in the distributions of population density by state?
















3
MATH 208 Final Exam    December 18th – 21st,  



Plot 1


20




15




10


IL

5




0




20




15


IN

10




5




0




20



count
15


MI

10




5









0




20




15


OH

10




5









0




20




15


WI

10




5




0




1e+02
1e+03
1e+04
1e+05


popdensity


grid.arrange(grobs=list(p2,p3),nrow=2,ncol=1)





Plot 2







state

1.0


IL
density



MI

0.5







IN




OH

0.0


WI






1e+02
1e+03
1e+04
1e+05



popdensity



Plot 3





state

WI

OH

MI

IN

IL

1e+02    1e+03    1e+04    1e+05

popdensity

CONTINUED ON NEXT PAGE


4
MATH 208 Final Exam    December 18th – 21st,  



    (c) [4 pts] Which plot(s) do you think best shows the association between state and population density? Which plot(s) do you think does not shows the association between state and population density as clearly? Explain your answer and reasoning in a few sentences.

    (d) [5 pts] Which of the following plots could also be used to assess the association between the popwhite and popblack variables? List all that apply (or say None if none would be appropriate).

A. 2-d density plot B. Barplot C. Boxplot D. 2-d histogram

We now would like to make plots to take a different look at the population variables. Unfortunately, the format of the midwest_modified data needs to be further changed so that we can use it in a ggplot.

    (e) [5 pts] Write a line of code that will create a new tibble converts the midwest_modified_new to “long” format where each row contains a population count for a specific racial group called Count, and the variable from where that count originated (e.g. popwhite) as well as the state, county, and Metro information for that population group. You should not include the columns for HighDens, inmetro or popdensity. The first 10 rows of the new tibble are below

midwest_modified_new %>% slice(1:10)


# A tibble: 10 x 5




county
state Metro
Race_Variable Count

<chr>
<chr> <chr>
<chr>
<dbl>
1
ADAMS
IL
NonMetro
popwhite
63917
2
ADAMS
IL
NonMetro
popblack
1702
3
ADAMS
IL
NonMetro
popamerindian
98
4
ADAMS
IL
NonMetro
popasian
249
5
ADAMS
IL
NonMetro
popother
124
6
ALEXANDER
IL
NonMetro
popwhite
7054
7
ALEXANDER
IL
NonMetro
popblack
3496
8
ALEXANDER
IL
NonMetro
popamerindian
19
9
ALEXANDER
IL
NonMetro
popasian
48
10
ALEXANDER
IL
NonMetro
popother
9

CONTINUED ON NEXT PAGE































5
MATH 208 Final Exam



Below is a figure along with the code (partially obscured) which generated it.

Plot f




3e+07





2e+07 

Count


1e+07






0e+00

Metro    NonMetro

Metro
December 18th – 21st,  














Race_Variable


popamerindian


popasian


popblack


popother


popwhite

ggplot(midwest_modified_new,aes(x=______,fill=______,y=______,


)) +

geom_XXXXXX(stat="identity") +    ggtitle("Plot f")    + theme_bw()

    (f) [5 pts] What are the missing geometry and aesthetics that generated the figure on the previous page (that is, what are the words that are missing in the code above for Plot f)?



    (g) [5 pts] Note that the plot in part (f) is a bit difficult to use because it contains the counts, rather than the relative proportions. Write a line of code (or lines of code) to create a new tibble called metro_race_summaries which contains each racial population count and proportion relative to the level of the Metro variable as below:

metro_race_summaries


    • A tibble: 10 x 4

    • Groups:   Metro [2]

Metro
Race_Variable Race_Count Proportion

<chr>
<chr>
<dbl>
<dbl>
1
Metro
popamerindian
99145
0.00296
2
Metro
popasian
538463
0.0161
3
Metro
popblack
4672825
0.140
4
Metro
popother
668449
0.0200
5
Metro
popwhite
27496337
0.821
6
NonMetro
popamerindian
50794
0.00595
7
NonMetro
popasian
34210
0.00401
8
NonMetro
popblack
144611
0.0169
9
NonMetro
popother
36402
0.00427
10
NonMetro
popwhite
8267706
0.969

6
MATH 208 Final Exam    December 18th – 21st,  



(h) [5 pts] Using the tibble from (g), write a line of code that created the barplot below.

Plot g



1.00

0.75
Proportion
0.50



0.25

0.00

Metro    NonMetro

Metro










Race_Variable


popamerindian


popasian


popblack


popother


popwhite

    (i) [5 pts] Based on the plot in part (h), would you conclude that there the population distribution of race varies between Metro and NonMetro areas? Explain your answer in a few sentences.

































7