Lab5: A/B Testing

library(MASS)
library(DT)
library(dplyr)

## 
## Attaching package: 'dplyr'

## The following object is masked from 'package:MASS':
## 
##     select

## The following objects are masked from 'package:stats':
## 
##     filter, lag

## The following objects are masked from 'package:base':
## 
##     intersect, setdiff, setequal, union

1 Matched Samples (Paired t-test)

1.1 Soal 18

In the built-in data set named immer, the barley yield in years 1931 and 1932 of the same field are recorded. The yield data are presented in the data frame columns Y1 and Y2. Assuming that the data in immer follows the normal distribution, find the 95% confidence interval estimate of the difference between the mean barley yields between years 1931 and 1932.

1.1.1 Data

DT::datatable(immer)

1.1.2 Estimate the difference between the means of matched samples using your textbook formula.

Y1 = immer$Y1
Y2 = immer$Y2
diff = Y1-Y2 

n = length(diff)
d = mean(diff)

sdiff = sqrt((1/(n-1))*sum(diff^2))

alpha = 1-0.95 ; alpha

## [1] 0.05

t = qt(1-(alpha/2), df = n-1)
E = c(-t,t) * (sdiff/sqrt(n)) ; E # nilai Error

## [1] -11.50642  11.50642

interval = round(d+E, digits = 2) ; interval

## [1]  4.41 27.42

Jadi didapatkan hasil selang kepercayaan dari rata-rata antara 4.41 dan 27.42.

\(4.41 ≤ μ ≤ 27.42\)

#Comparison Proportions

1.2 Soal 19

In the built-in data set named quine, children from an Australian town is classified by ethnic background, gender, age, learning status and the number of days absent from school. In effect, the data frame column Eth indicates whether the student is Aboriginal or Not (“A” or “N”), and the column Sex indicates Male or Female (“M” or “F”). Assuming that the data in quine follows the normal distribution, find the 95% confidence interval estimate of the difference between the female proportion of Aboriginal students and the female proportion of Non-Aboriginal students, each within their own ethnic group.

In R, we can tally the student ethnicity against the gender with the table function. As the result shows, within the Aboriginal student population, 38 students are female. Whereas within the Non-Aboriginal student population, 42 are female.

1.2.1 DATA

DT::datatable(quine)

1.2.2 Estimate the difference between two population proportions using your textbook formula.

table(quine$Eth, quine$Sex)

##    
##      F  M
##   A 38 31
##   N 42 35

data = quine%>%
  count(Eth, Sex)
data = data.frame(
  "-" = c("A" , "N" , "TOTAL"),
  "F" = c(38,42, 38+42),
  "M" = c(31,35, 31+35),
  "TOTAL" = c(38+31, 42+35, 38+31+42+35)
)
data

##      X.  F  M TOTAL
## 1     A 38 31    69
## 2     N 42 35    77
## 3 TOTAL 80 66   146

esti1 = 38/69
n1 = 69
esti2 = 42/77
n2 = 77
alpha = .05
estimate = esti1-esti2
x = qnorm(0.975) * sqrt(((esti1*(1-esti1))/n1))+sqrt(((esti2*(1-esti2))/n2))

CI = round(estimate + c(-x,x), digits = 3) ; CI

## [1] -0.169  0.179

Selang kepercayaannya adalah [-0.169, 0.179] atau [-16.9%, 17.9%]

prop.test(table(quine$Eth, quine$Sex), correct = FALSE)

## 
##  2-sample test for equality of proportions without continuity correction
## 
## data:  table(quine$Eth, quine$Sex)
## X-squared = 0.0040803, df = 1, p-value = 0.9491
## alternative hypothesis: two.sided
## 95 percent confidence interval:
##  -0.1564218  0.1669620
## sample estimates:
##    prop 1    prop 2 
## 0.5507246 0.5454545