Nama Anggota Kelompok:

Maulana Diki Wicaksono (23031030001)
Nisrina Aisyah (23031030041)
Divna Widyastuti (23031030006)
Siti Zaniah Nur Hidayah (23031030039)
Nafisa Salsabila (23031030048)

SOAL 1

The table here shows the ANOVA table for a regression analysis of y = the selling price (in thousands of dollars) and x = the size of house (in thousands of square feet). The prediction equation is ybar = 9.2 + 77(x).

Diketahui Tabel Anova

library(readxl)
  data1<-read.csv("C:/Users/ASUS/Documents/UNY/SEM 2/ANALISIS REGRESI/Tugas AnReg/soal1.csv")
  data1<- data.frame(data1)
    head(data1)

##       Source DF     SS MS  F
## 1 Regression  1 500000 NA NA
## 2      Error 35 300000 NA NA
## 3      Total 36 800000 NA NA

Jawab:

- Menghitung Mean Square Regression (MSR)

MSR = SSR/DFR

  MSR <- data1[1,3]/data1[1,2]
    MSR

## [1] 5e+05

- Menghitung Mean Square Error (MSE)

MSE = SSE/DFE

  MSE <- data1[2,3]/data1[2,2]
    MSE

## [1] 8571.429

- Menghitung F-value

  F_value <- sum(MSR/MSE)
    F_value

## [1] 58.33333

- Jawaban Akhir Tabel Anova

  source <- c("Regression", "Error", "Total")
  DF <- c(1,35,36)
  SS = c(500000,300000,800000)
  MS <- c(MSR,MSE, " ")
  F <- c(F_value, " ", " ")
  data1 <- data.frame(source, DF, SS, MS, F)
    data1

##       source DF    SS               MS                F
## 1 Regression  1 5e+05            5e+05 58.3333333333333
## 2      Error 35 3e+05 8571.42857142857                 
## 3      Total 36 8e+05

SOAL 2

The table here shows the ANOVA table for a regression analysis of y = the selling price (in thousands of dollars) and x = the size of house (in thousands of square feet). The prediction equation is ybar =9.2 + 77(x)

- Tabel ANOVA

library(readxl)
  data2<-read.csv("C:/Users/ASUS/Documents/UNY/SEM 2/ANALISIS REGRESI/Tugas AnReg/soal2.csv")
  data2<-data.frame(data2)
    head(data2)

##       Source DF     SS     MS      F  p
## 1 Regression  1 182220 182220 135.07  0
## 2      Error 98 132213   1349     NA NA
## 3      Total 99 314433     NA     NA NA

a. What was the sample size?

Jawab:

#DF Total = n-1
  sample_size <- sum((data2[3,2]+1))
    sample_size

## [1] 100

b. The sample mean house size was 1.53 thousand square feet. What was the sample mean selling price? (Hint: What does ybar equal when x=xbar ?)

Jawab:

# The sample mean selling price
  # y = the selling price (in thousands of dollars)
  # x = the size of house (in thousands square feet)
  # yhat equal when x = xbar, yhat = y
    x <- 1.53
    yhat <- 9.2 + 77*x
      yhat

## [1] 127.01

SOAL 3

Create a regression equation based on the data in this link give interpretation for the regression equation, determine the determination coefficient, correlation, and do the hypothesis testing for correlation.

Input Data

  library(readxl)
    data<-read.csv("C:/Users/ASUS/Documents/UNY/SEM 2/ANALISIS REGRESI/Tugas AnReg/data_tugas3.csv")
      head(data)

##   Height Weight
## 1   1.47  52.21
## 2   1.50  53.12
## 3   1.52  54.48
## 4   1.55  55.84
## 5   1.57  57.20
## 6   1.60  58.57

Jawab:

- Eksplorasi Data

  plot(data$Height, data$Weight,xlab="Height",ylab="Weight",pch=16)

Persamaan Regresi

- Persamaan Regresi dengan Metode Kuadrat Terkecil (MKT) Manual

  data$xdif <-   data$Height-mean(data$Height)
  data$ydif <- data$Weight-mean(data$Weight)
  data$crp <- data$xdif*data$ydif
  data$xsq <- data$xdif^2
#estimator b0 dan b1
  b1 <- sum(data$crp)/sum(data$xsq)
  b1

## [1] 61.27219

#Parameter Estimates
  b0 <- mean(data$Weight) - b1 * mean(data$Height)
  b0

## [1] -39.06196

- Persamaan Regresi dengan R function

model1<-lm(Weight~Height,data=data)
summary(model1)

## 
## Call:
## lm(formula = Weight ~ Height, data = data)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.88171 -0.64484 -0.06993  0.34095  1.39385 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  -39.062      2.938  -13.29 6.05e-09 ***
## Height        61.272      1.776   34.50 3.60e-14 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.7591 on 13 degrees of freedom
## Multiple R-squared:  0.9892, Adjusted R-squared:  0.9884 
## F-statistic:  1190 on 1 and 13 DF,  p-value: 3.604e-14

- Plot Regresi

  plot(data$Height, data$Weight,xlab="Height",ylab="Weight",pch=16)
  abline(model1)

- Koefisien Determinasi

  JKG<-sum((data$Weight-model1$fitted.values)^2)
    JKG

## [1] 7.490558

  JKR<-sum(((model1$fitted.values-mean(data$Weight))^2))
    JKR

## [1] 685.8821

  JKT<-sum((data$Weight-mean(data$Weight))^2)
    JKT

## [1] 693.3726

#koefisien determinasi
  r_sqr<-(JKT-JKG)/JKT
    r_sqr

## [1] 0.9891969

- Korelasi

#koefisien korelasi
  sqrt(r_sqr)

## [1] 0.9945838

ANOVA

  anova(model1)

## Analysis of Variance Table
## 
## Response: Weight
##           Df Sum Sq Mean Sq F value    Pr(>F)    
## Height     1 685.88  685.88  1190.4 3.604e-14 ***
## Residuals 13   7.49    0.58                      
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

#koefisien korelasi cara 1
  x1_x1bar<-data$Height-mean(data$Height)
  x2_x2bar<-data$Weight-mean(data$Weight)
  A<-sum(x1_x1bar*x2_x2bar)
  varx1<-sum((x1_x1bar)^2)
  varx2<-sum((x2_x2bar)^2)
  B<-sqrt(varx1*varx2)
  corr<-A/B
    corr

## [1] 0.9945838

#koefisien korelasi cara 2
  cor(data$Height,data$Weight)

## [1] 0.9945838

- Uji Signifikansi Korelasi

#Pearson Correlation test, alpha 1%
  cor.test( ~ Height + Weight, data=data, method = "pearson", continuity = FALSE, conf.level = 0.99)

## 
##  Pearson's product-moment correlation
## 
## data:  Height and Weight
## t = 34.502, df = 13, p-value = 3.604e-14
## alternative hypothesis: true correlation is not equal to 0
## 99 percent confidence interval:
##  0.9762562 0.9987733
## sample estimates:
##       cor 
## 0.9945838

HIPOTESIS

- Hipotesis Nol

Menyatakan bahwa tidak ada hubungan antara tinggi badan dan berat badan seseorang, atau dengan kata lain, korelasinya adalah nol. H0 : p = 0

- Hipotesis Alternatif

Menyatakan bahwa ada hubungan antara tinggi badan dan berat badan seseorang, atau dengan kata lain, korelasinya adalah tidak nol. H0 : p ≠ 0

Daerah Penolakan

H0 ditolak jika p-value < tingkat signifikansi (alpha)

KESIMPULAN

Dengan nilai koefisien korelasi sebesar 0.9945838, menunjukkan hubungan positif yang sangat kuat antara tinggi dan berat badan.

Hasil uji signifikansi menunjukkan bahwa korelasi ini secara statistik signifikan dengan p-value sebesar 3.604e-14. Dimana p-value < tingkat signifikansi (0.01), Sehingga :

hipotesis nol(H0) dapat ditolak.

Dengan Demikian, dapat disimpulkan bahwa terdapat korelasi antara tinggi badan dan berat badan.

SOAL 4

Susun tabel ANOVA, kemudian hitung koefisien determinasi dan koefisien korelasi dari data berikut, dan jelaskan serta lakukan pengujian!

Input Data

  library(readxl)
    data4<-read.csv("C:/Users/ASUS/Documents/UNY/SEM 2/ANALISIS REGRESI/Tugas AnReg/data4.csv")
      head(data4)

##   Umur Massa.Otot
## 1   71         82
## 2   64         91
## 3   43        100
## 4   67         68
## 5   56         87
## 6   73         73

Jawab:

- Eksplorasi Data

  plot(data4$Umur, data4$Massa.Otot,xlab="Umur",ylab="Massa.Otot",pch=16)

Persamaan Regresi

- Persamaan Regresi dengan MKT Manual

data4$xdif <-   data4$Umur-mean(data4$Umur)
  data4$ydif <- data4$Massa.Otot-mean(data4$Massa.Otot)
  data4$crp <- data4$xdif*data4$ydif
  data4$xsq <- data4$xdif^2
#estimator b0 dan b1
  b1 <- sum(data4$crp)/sum(data4$xsq)
   b1

## [1] -1.023589

#Parameter Estimates
  b0 <- mean(data4$Massa.Otot) - b1 * mean(data4$Umur)
    b0

## [1] 148.0507

- Persamaan Regresi dengan R function

  model1<-lm(Umur~Massa.Otot,data=data4)
  summary(model1)

## 
## Call:
## lm(formula = Umur ~ Massa.Otot, data = data4)
## 
## Residuals:
##    Min     1Q Median     3Q    Max 
## -9.193 -6.191  1.617  3.946 11.470 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 117.5935    10.6411  11.051 2.67e-08 ***
## Massa.Otot   -0.6632     0.1219  -5.439 8.72e-05 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 6.716 on 14 degrees of freedom
## Multiple R-squared:  0.6788, Adjusted R-squared:  0.6559 
## F-statistic: 29.59 on 1 and 14 DF,  p-value: 8.721e-05

- Plot Regresi

  plot(data4$Umur, data4$Massa.Otot,xlab="Umur",ylab="Massa.Otot",pch=16)
  abline(model1)

- Koefisien Determinasi

 JKG<-sum((data4$Massa.Otot-model1$fitted.values)^2)
    JKG

## [1] 19002.54

JKR<-sum(((model1$fitted.values-mean(data4$Massa.Otot))^2))
    JKR

## [1] 11943.48

  JKT<-sum((data4$Massa.Otot-mean(data4$Massa.Otot))^2)
    JKT

## [1] 3034.438

#koefisien determinasi
  r_sqr<-(JKT-JKG)/JKT
    r_sqr

## [1] -5.262295

- ANOVA

anova(model1)

## Analysis of Variance Table
## 
## Response: Umur
##            Df  Sum Sq Mean Sq F value    Pr(>F)    
## Massa.Otot  1 1334.48  1334.5  29.587 8.721e-05 ***
## Residuals  14  631.46    45.1                      
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

#koefisien korelasi cara 1
  x1_x1bar<-data4$Umur-mean(data4$Umur)
  x2_x2bar<-data4$Massa.Otot-mean(data4$Massa.Otot)
  A<-sum(x1_x1bar*x2_x2bar)
  varx1<-sum((x1_x1bar)^2)
  varx2<-sum((x2_x2bar)^2)
  B<-sqrt(varx1*varx2)
  corr<-A/B
    corr

## [1] -0.8238943

#koefisien korelasi cara 2
  cor(data4$Umur,data4$Massa.Otot)

## [1] -0.8238943

- Uji Signifikansi Korelasi

#Pearson Correlation test, alpha 1%
  cor.test( ~ Umur + Massa.Otot, data=data4, method = "pearson", continuity = FALSE, conf.level = 0.99)

## 
##  Pearson's product-moment correlation
## 
## data:  Umur and Massa.Otot
## t = -5.4394, df = 14, p-value = 8.721e-05
## alternative hypothesis: true correlation is not equal to 0
## 99 percent confidence interval:
##  -0.9547784 -0.4255220
## sample estimates:
##        cor 
## -0.8238943

HIPOTESIS

- Hipotesis Nol

Menyatakan bahwa tidak ada hubungan antara umur dan massa otot seseorang (korelasinya adalah nol). H0 : p = 0

- Hipotesis Alternatif

Menyatakan bahwa ada hubungan antara umur dan massa otot seseorang (korelasinya adalah tidak nol). H0 : p ≠ 0

Daerah Penolakan

H0 ditolak jika p-value < tingkat signifikansi (alpha)

KESIMPULAN

Dari perhitungan R didapat p-value = (8.721e-05), dimana hasil p-value < tingkat signifikansi yang umum digunakan (0.05). Sehingga :

hipotesis nol(H0) dapat ditolak.

Dengan Demikian, dapat disimpulkan bahwa terdapat korelasi antara Umur dan massa otot.

Tugas Analisis Regresi

Kelompok 1

2024-02-28

Nama Anggota Kelompok:

SOAL 1

The table here shows the ANOVA table for a regression analysis of y = the selling price (in thousands of dollars) and x = the size of house (in thousands of square feet). The prediction equation is ybar = 9.2 + 77(x).

Diketahui Tabel Anova

Jawab:

- Menghitung Mean Square Regression (MSR)

MSR = SSR/DFR

- Menghitung Mean Square Error (MSE)

MSE = SSE/DFE

- Menghitung F-value

- Jawaban Akhir Tabel Anova

SOAL 2

The table here shows the ANOVA table for a regression analysis of y = the selling price (in thousands of dollars) and x = the size of house (in thousands of square feet). The prediction equation is ybar =9.2 + 77(x)

- Tabel ANOVA

a. What was the sample size?

Jawab:

b. The sample mean house size was 1.53 thousand square feet. What was the sample mean selling price? (Hint: What does ybar equal when x=xbar ?)

Jawab:

SOAL 3

Create a regression equation based on the data in this link give interpretation for the regression equation, determine the determination coefficient, correlation, and do the hypothesis testing for correlation.

Input Data

Jawab:

- Eksplorasi Data

Persamaan Regresi

- Persamaan Regresi dengan Metode Kuadrat Terkecil (MKT) Manual

- Persamaan Regresi dengan R function

- Plot Regresi

- Koefisien Determinasi

- Korelasi

ANOVA

- Uji Signifikansi Korelasi

HIPOTESIS

- Hipotesis Nol

- Hipotesis Alternatif

Daerah Penolakan

KESIMPULAN

SOAL 4

Susun tabel ANOVA, kemudian hitung koefisien determinasi dan koefisien korelasi dari data berikut, dan jelaskan serta lakukan pengujian!

Input Data

Jawab:

- Eksplorasi Data

Persamaan Regresi

- Persamaan Regresi dengan MKT Manual

- Persamaan Regresi dengan R function

- Plot Regresi

- Koefisien Determinasi

- ANOVA

- Uji Signifikansi Korelasi

HIPOTESIS

- Hipotesis Nol

- Hipotesis Alternatif

Daerah Penolakan

KESIMPULAN

Sekian, Terima Kasih :)