Q1

The data set UCBAdmissions is a 3-way table of frequencies classified by Admit, Gender, and Dept.

  1. Find the total number of cases contained in this table.
data("UCBAdmissions")
View(UCBAdmissions)
summary(UCBAdmissions)
## Number of cases in table: 4526 
## Number of factors: 3 
## Test for independence of all factors:
##  Chisq = 2000.3, df = 16, p-value = 0

The number of cases in this table is 4526.

  1. For each department, find the total number of applicants.
table1<-UCBAdmissions%>%
  as.data.frame()%>%
  group_by(Dept)%>%
   summarize(sum1=sum(Freq))
table1
## # A tibble: 6 x 2
##   Dept   sum1
##   <fct> <dbl>
## 1 A      933.
## 2 B      585.
## 3 C      918.
## 4 D      792.
## 5 E      584.
## 6 F      714.
  1. For each department, find the overall proportion of applicants who were admitted.
table2<-UCBAdmissions%>%
  as.data.frame()%>%
  group_by(Dept,Admit)%>%
   summarize(sum2=sum(Freq)) %>%
   filter(Admit=="Admitted")

table2$admission_rate<-100*table2$sum2/(table1$sum1)
table2
## # A tibble: 6 x 4
## # Groups:   Dept [6]
##   Dept  Admit     sum2 admission_rate
##   <fct> <fct>    <dbl>          <dbl>
## 1 A     Admitted  601.          64.4 
## 2 B     Admitted  370.          63.2 
## 3 C     Admitted  322.          35.1 
## 4 D     Admitted  269.          34.0 
## 5 E     Admitted  147.          25.2 
## 6 F     Admitted   46.           6.44
  1. Construct a tabular display of department (rows) and gender (columns), showing the proportion of applicants in each cell who were admitted relative to the total applicants in that cell.
table3<-t(percentages(UCBAdmissions,which="Gender",by="Dept"))
table3
##     Gender
## Dept      Male    Female
##    A 88.424437 11.575563
##    B 95.726496  4.273504
##    C 35.403050 64.596950
##    D 52.651515 47.348485
##    E 32.705479 67.294521
##    F 52.240896 47.759104

Q2

This two-way table classifies all 20 × 19 = 380 games by the joint outcome (Home, Away), the number of goals scored by the Home and Away teams. The value 4 in this table actually represents 4 or more goals.

  1. Verify that the total number of games represented in this table is 380.
data("UKSoccer",package = "vcd")
ftable(UKSoccer)
##      Away  0  1  2  3  4
## Home                    
## 0         27 29 10  8  2
## 1         59 53 14 12  4
## 2         28 32 14 12  4
## 3         19 14  7  4  1
## 4          7  8 10  2  0
summary(UKSoccer)
## Number of cases in table: 380 
## Number of factors: 2 
## Test for independence of all factors:
##  Chisq = 18.699, df = 16, p-value = 0.2846
##  Chi-squared approximation may be incorrect
  1. Find the marginal total of the number of goals scored by each of the home and away teams.
table4<-addmargins(UKSoccer)
table4
##      Away
## Home    0   1   2   3   4 Sum
##   0    27  29  10   8   2  76
##   1    59  53  14  12   4 142
##   2    28  32  14  12   4  90
##   3    19  14   7   4   1  45
##   4     7   8  10   2   0  27
##   Sum 140 136  55  38  11 380
  1. Express each of the marginal totals as proportions.
vec1<-t(prop.table(table4[1:5,6]))

vec2<-prop.table(table4[6,1:5])

table5<-table4%>%
  as.data.frame()%>%
  filter(Home=="Sum") %>%
  mutate(prec_away=Freq/380)
table5
##   Home Away Freq  prec_away
## 1  Sum    0  140 0.36842105
## 2  Sum    1  136 0.35789474
## 3  Sum    2   55 0.14473684
## 4  Sum    3   38 0.10000000
## 5  Sum    4   11 0.02894737
## 6  Sum  Sum  380 1.00000000
table6<-table4%>%
  as.data.frame()%>%
  filter(Away=="Sum") %>%
  mutate(prec_Home=Freq/380)
table6
##   Home Away Freq  prec_Home
## 1    0  Sum   76 0.20000000
## 2    1  Sum  142 0.37368421
## 3    2  Sum   90 0.23684211
## 4    3  Sum   45 0.11842105
## 5    4  Sum   27 0.07105263
## 6  Sum  Sum  380 1.00000000

The perc_home calculates the percentage of the frequency of the away team’s scores. For example, away team score 1 in 136 games, which is 35.79% percentage of the total 380 games. The perc_away calculates the percentage of the frequency of the home team’s score. For example, home team score 0 in 76 games, which is 20% of the total 380 games.

  1. Comment on the distribution of the numbers of home-team and away-team goals. Is there any evidence that home teams score more goals on average?
homesum<-sum(margin.table(UKSoccer,1)*c(0,1,2,3,4))
homesum
## [1] 565
homemean<-homesum/380
homemean
## [1] 1.486842
awaysum<-sum(margin.table(UKSoccer,2)*c(0,1,2,3,4))
awaysum
## [1] 404
awaymean<-awaysum/380
awaymean
## [1] 1.063158

The results shows that the total score that home team got is 565 and average score per team is 1.49. Away team score in total 404 goals, on leverage score 1.06 each game, less than home team