Assignment 2
Anshumaan
************************************************************************
# Q1a. Write R code to generate the correlation matrix for the given continuous variables {"Price", "AdvancedBookingDays", "FlyingMinutes", "Capacity", "SeatPitch", "SeatWidth"}
matrix1a <- cor(price1[,c("Price", "AdvancedBookingDays", "FlyingMinutes", "Capacity", "SeatPitch", "SeatWidth")])
# round upto 2 decimal places
round(matrix1a, 2) ## Price AdvancedBookingDays FlyingMinutes Capacity
## Price 1.00 -0.01 -0.02 -0.03
## AdvancedBookingDays -0.01 1.00 0.01 -0.01
## FlyingMinutes -0.02 0.01 1.00 -0.32
## Capacity -0.03 -0.01 -0.32 1.00
## SeatPitch 0.07 -0.01 -0.03 0.51
## SeatWidth -0.06 0.05 -0.18 0.45
## SeatPitch SeatWidth
## Price 0.07 -0.06
## AdvancedBookingDays -0.01 0.05
## FlyingMinutes -0.03 -0.18
## Capacity 0.51 0.45
## SeatPitch 1.00 0.32
## SeatWidth 0.32 1.00************************************************************************
#Q1b. Write R code to generate the correlation matrix, along with their significance values, for the given continuous variables {"Price", "AdvancedBookingDays", "FlyingMinutes", "Capacity", "SeatPitch", "SeatWidth"}
library(Hmisc)
price2 <- price1[,c("Price", "AdvancedBookingDays", "FlyingMinutes", "Capacity", "SeatPitch", "SeatWidth")]
rcorr(as.matrix(price2), type="pearson")************************************************************************
#Q1c. Write R code to visualize the correlation matrix in Q1b.
library(corrgram)
price2 <- price1[,c("Price", "AdvancedBookingDays", "FlyingMinutes", "Capacity", "SeatPitch", "SeatWidth")]
corrgram(as.matrix(price2), order=TRUE, lower.panel=panel.conf,
upper.panel=panel.pie, text.panel=panel.txt,
main="Visualization of correlation")
#************************************************************************
#Q1d. Write R code to generate the corrgram using PerformanceAnalytics package
library("PerformanceAnalytics")## Loading required package: xts## Loading required package: zoo##
## Attaching package: 'zoo'## The following objects are masked from 'package:base':
##
## as.Date, as.Date.numeric##
## Attaching package: 'PerformanceAnalytics'## The following object is masked from 'package:graphics':
##
## legendchart.Correlation(price2, histogram = TRUE, pch=19)
#Q2a. Test whether the ticket prices of Mumbai to Delhi flights are more than INR 5000.
# H0: BOM - DEL flights ticket price is equal to INR 5000
# Alternate Hypothesis: BOM - DEL flights ticket price is more than INR 5000
attach(price1)
# Inspect structure of data
head(price1)## FlightNumber Airline DepartureCityCode ArrivalCityCode DepartureTime
## 1 9W 313 Jet DEL BOM 225
## 2 9W 339 Jet BOM DEL 300
## 3 SG 161 Spice Jet DEL BOM 350
## 4 6E 171 IndiGo DEL BOM 455
## 5 SG 160 Spice Jet BOM DEL 555
## 6 9W 762 Jet BOM DEL 605
## ArrivalTime Departure FlyingMinutes Aircraft PlaneModel Capacity
## 1 435 AM 130 Boeing 738 156
## 2 505 AM 125 Boeing 738 156
## 3 605 AM 135 Boeing 738 189
## 4 710 AM 135 Airbus A320 180
## 5 805 AM 130 Boeing 738 189
## 6 815 AM 130 Boeing 738 156
## SeatPitch SeatWidth DataCollectionDate DateDeparture IsWeekend Price
## 1 30 17 Sep 13 2018 Nov 6 2018 No 4051
## 2 30 17 Sep 15 2018 Nov 6 2018 No 11587
## 3 29 17 Sep 19 2018 Nov 6 2018 No 3977
## 4 30 18 Sep 8 2018 Nov 6 2018 No 4234
## 5 29 17 Sep 19 2018 Nov 6 2018 No 6837
## 6 30 17 Sep 15 2018 Nov 6 2018 No 6518
## AdvancedBookingDays IsDiwali DayBeforeDiwali DayAfterDiwali
## 1 54 1 1 0
## 2 52 1 1 0
## 3 48 1 1 0
## 4 59 1 1 0
## 5 48 1 1 0
## 6 52 1 1 0
## MetroDeparture MetroArrival MarketShare LoadFactor
## 1 1 1 15.4 83.32
## 2 1 1 15.4 83.32
## 3 1 1 13.2 94.06
## 4 1 1 39.6 87.20
## 5 1 1 13.2 94.06
## 6 1 1 15.4 83.32price_Mum_Del_data <- price1$Price[ which( price1$DepartureCityCode =="BOM" & price1$ArrivalCityCode =="DEL" )]
# Perform a t test on ticket price from Bom to Del (one-tailed test)
t.test(price_Mum_Del_data, mu=5000, alternative = "greater")##
## One Sample t-test
##
## data: price_Mum_Del_data
## t = 6.0784, df = 129, p-value = 6.385e-09
## alternative hypothesis: true mean is greater than 5000
## 95 percent confidence interval:
## 5910.787 Inf
## sample estimates:
## mean of x
## 6252.054# Result of Independent t-test
# Based on the above output of the t-test, we can REJECT the hypothesis (p < 0.001) as the p value is less than 0.001. The t test showed there the ticket prices of Mumbai (BOM) to Delhi (DEL) flights is not equal to INR 5000 and the mean value (6252.054) is greater than INR 5000. ************************************************************************
# Q2b. Test whether the ticket prices of morning flights are greater than the afternoon flights
# H0: Ticket price for morning flights are equal to Ticket price for afternoon flights
# Alternate Hypothesis : The ticket prices of morning flights are greater than the afternoon flights
# Generate descriptive statistics for each group
aggregate(Price~Departure, data=price1, FUN = mean)## Departure Price
## 1 AM 5598.893
## 2 PM 5140.610# Generate box plot for each group
boxplot(Price~Departure, main = "Average Ticket Prices", xlab = "Departure Time",col = (c("green","blue")), ylab = "Average Ticket Price")
# Perform a t test on the transformed variable
t.test(Price~Departure, var.equal = TRUE, alternative="greater")##
## Two Sample t-test
##
## data: Price by Departure
## t = 1.6706, df = 303, p-value = 0.04791
## alternative hypothesis: true difference in means is greater than 0
## 95 percent confidence interval:
## 5.691872 Inf
## sample estimates:
## mean in group AM mean in group PM
## 5598.893 5140.610#Result of Independent t-test
# Based on the above output of the t-test, the p-value is greater then 0.001, hence we have no evidence for rejecting the null hypothesis. There is no significance difference in the ticket prices of morning flights compared to afternoon flights. ************************************************************************
# Q2c. Test whether the ticket prices around Diwali is more compared to non-Diwali ticket prices
# H0: Ticket Prices during Diwali is equal to non-Diwali ticket prices
# Alternate Hypothesis: Ticket Prices during Diwali is more compared to non-Diwali ticket prices
# Generate descriptive statistics for each group
aggregate(Price~IsDiwali, data=price1, FUN = mean)## IsDiwali Price
## 1 0 5063.810
## 2 1 5897.479# Generate box plot for each group
boxplot(Price~IsDiwali, main = "Average Ticket Prices", xlab = "Diwali Time",col = (c("green","blue")), ylab = "Average Ticket Price")
# Perform a t test on the transformed variable
t.test(Price~IsDiwali,var.equal = TRUE, alternative="less")##
## Two Sample t-test
##
## data: Price by IsDiwali
## t = -3.022, df = 303, p-value = 0.001363
## alternative hypothesis: true difference in means is less than 0
## 95 percent confidence interval:
## -Inf -378.5134
## sample estimates:
## mean in group 0 mean in group 1
## 5063.810 5897.479#t.test(airlineData.df[airlineData.df$IsDiwali==1,]$Price, airlineData.df[airlineData.df$IsDiwali==0,]$Price, var.equal = TRUE, alternative="greater")
# Result of Independent t-test
# Based on the output of the t-test, we dont have evidence of rejecting the null hypothesis (p < 0.001) since the p value is slightly greater than 0.001. ************************************************************************
# Q2d. Test whether the ticket prices on Air India flights are greater than IndiGo flights
# H0: Ticket Prices (Air India) = Ticket Prices (Indigo)
# H1: Ticket Prices (Air India) > Ticket Prices (Indigo)
# Generate descriptive statistics for each group
aggregate(Price~Airline, data=price1, FUN = mean)## Airline Price
## 1 Air India 6335.000
## 2 IndiGo 4879.525
## 3 Jet 5496.146
## 4 Spice Jet 5094.850# Generate box plot for each group
boxplot(Price~Airline, main = "Average Ticket Prices", xlab = "Airline",col = (c("green","blue")), ylab = "Average Ticket Price")
# Perform a t test on the transformed variable
AI_data = price1$Price[price1$Airline == 'Air India']
Indigo_data = price1$Price[price1$Airline == 'IndiGo']
# Number of observations
length(AI_data)## [1] 41length(Indigo_data)## [1] 80t.test(AI_data, Indigo_data, paired=FALSE, var.equal = TRUE, alternative="greater")##
## Two Sample t-test
##
## data: AI_data and Indigo_data
## t = 2.6396, df = 119, p-value = 0.004705
## alternative hypothesis: true difference in means is greater than 0
## 95 percent confidence interval:
## 541.4039 Inf
## sample estimates:
## mean of x mean of y
## 6335.000 4879.525# Result of Independent t-test
# Based on the above output of the t-test,we dont have evidence of rejecting the null hypothesis (p < 0.001) since the p value is slightly greater than 0.001. ************************************************************************
# QUESTION 3
# Q3a. Run a simple linear regression of airline ticket Price on the Advanced Booking Days. Write R code to output the summary of the model
# Independent variable (say x)
xAdvBookDays <- price1$AdvancedBookingDays
# Dependent Variable (say y)
yPrice <- price1$Price
# Plot Advance Booking Days vs Price Graph
plot(xAdvBookDays, yPrice, xlab="Advance Booking Days", ylab="Price (INR)")
scatter.smooth(xAdvBookDays, yPrice)
# Simple Linear Regression Model
fit <- lm(yPrice ~ xAdvBookDays, data = price1)
# Beta Coefficients
fit$coefficients## (Intercept) xAdvBookDays
## 5422.9589348 -0.9830394#***************** Model: price = 5422.959 - 0.983 ??? Advance Booking Days ****************
lines(xAdvBookDays, (5422.9589348 - 0.9830394 * xAdvBookDays), col = 3)
# Confidence Interval (95%)
confint(fit)## 2.5 % 97.5 %
## (Intercept) 4981.18757 5864.73030
## xAdvBookDays -13.09224 11.12616# Predicted Values
fitted(fit) ## 1 2 3 4 5 6 7 8
## 5369.875 5371.841 5375.773 5364.960 5375.773 5371.841 5375.773 5364.960
## 9 10 11 12 13 14 15 16
## 5375.773 5371.841 5369.875 5375.773 5364.960 5371.841 5375.773 5375.773
## 17 18 19 20 21 22 23 24
## 5371.841 5369.875 5375.773 5364.960 5371.841 5369.875 5373.807 5375.773
## 25 26 27 28 29 30 31 32
## 5364.960 5375.773 5375.773 5369.875 5373.807 5375.773 5364.960 5373.807
## 33 34 35 36 37 38 39 40
## 5364.960 5369.875 5375.773 5370.858 5373.807 5364.960 5366.926 5370.858
## 41 42 43 44 45 46 47 48
## 5373.807 5375.773 5366.926 5366.926 5370.858 5373.807 5370.858 5366.926
## 49 50 51 52 53 54 55 56
## 5375.773 5370.858 5366.926 5370.858 5366.926 5371.841 5375.773 5366.926
## 57 58 59 60 61 62 63 64
## 5371.841 5366.926 5371.841 5367.909 5369.875 5373.807 5362.994 5373.807
## 65 66 67 68 69 70 71 72
## 5369.875 5373.807 5362.994 5373.807 5369.875 5367.909 5373.807 5362.994
## 73 74 75 76 77 78 79 80
## 5369.875 5373.807 5373.807 5369.875 5367.909 5373.807 5362.994 5369.875
## 81 82 83 84 85 86 87 88
## 5367.909 5371.841 5373.807 5362.994 5373.807 5373.807 5367.909 5371.841
## 89 90 91 92 93 94 95 96
## 5373.807 5362.994 5371.841 5362.994 5367.909 5373.807 5368.892 5371.841
## 97 98 99 100 101 102 103 104
## 5362.994 5364.960 5368.892 5371.841 5373.807 5364.960 5364.960 5368.892
## 105 106 107 108 109 110 111 112
## 5371.841 5368.892 5364.960 5371.841 5373.807 5368.892 5364.960 5368.892
## 113 114 115 116 117 118 119 120
## 5364.960 5371.841 5369.875 5373.807 5364.960 5369.875 5364.960 5371.841
## 121 122 123 124 125 126 127 128
## 5369.875 5393.468 5393.468 5393.468 5393.468 5393.468 5393.468 5393.468
## 129 130 131 132 133 134 135 136
## 5393.468 5393.468 5393.468 5393.468 5393.468 5393.468 5393.468 5393.468
## 137 138 139 140 141 142 143 144
## 5393.468 5393.468 5393.468 5393.468 5393.468 5393.468 5393.468 5393.468
## 145 146 147 148 149 150 151 152
## 5393.468 5393.468 5393.468 5393.468 5393.468 5393.468 5393.468 5393.468
## 153 154 155 156 157 158 159 160
## 5393.468 5393.468 5393.468 5393.468 5393.468 5393.468 5393.468 5393.468
## 161 162 163 164 165 166 167 168
## 5393.468 5393.468 5393.468 5393.468 5393.468 5393.468 5393.468 5393.468
## 169 170 171 172 173 174 175 176
## 5393.468 5393.468 5393.468 5393.468 5393.468 5393.468 5393.468 5393.468
## 177 178 179 180 181 182 183 184
## 5393.468 5393.468 5393.468 5393.468 5420.993 5420.993 5420.993 5420.993
## 185 186 187 188 189 190 191 192
## 5420.993 5420.993 5420.993 5420.993 5420.993 5420.993 5420.993 5420.993
## 193 194 195 196 197 198 199 200
## 5420.993 5420.993 5420.993 5420.993 5420.993 5416.078 5416.078 5420.993
## 201 202 203 204 205 206 207 208
## 5416.078 5420.993 5416.078 5420.993 5416.078 5420.993 5416.078 5416.078
## 209 210 211 212 213 214 215 216
## 5420.993 5416.078 5420.993 5420.993 5420.993 5420.993 5420.993 5420.993
## 217 218 219 220 221 222 223 224
## 5420.993 5420.993 5420.993 5420.993 5420.993 5420.993 5416.078 5416.078
## 225 226 227 228 229 230 231 232
## 5416.078 5416.078 5416.078 5416.078 5420.993 5416.078 5420.993 5416.078
## 233 234 235 236 237 238 239 240
## 5420.993 5420.993 5420.993 5420.993 5420.993 5420.993 5420.993 5420.993
## 241 242 243 244 245 246 247 248
## 5420.993 5420.993 5416.078 5416.078 5416.078 5416.078 5416.078 5416.078
## 249 250 251 252 253 254 255 256
## 5420.993 5420.993 5420.993 5420.993 5420.993 5420.993 5420.993 5420.993
## 257 258 259 260 261 262 263 264
## 5420.993 5420.993 5420.993 5420.993 5420.993 5416.078 5416.078 5420.993
## 265 266 267 268 269 270 271 272
## 5416.078 5416.078 5420.993 5416.078 5416.078 5420.993 5420.993 5416.078
## 273 274 275 276 277 278 279 280
## 5416.078 5416.078 5416.078 5416.078 5416.078 5416.078 5416.078 5416.078
## 281 282 283 284 285 286 287 288
## 5416.078 5416.078 5416.078 5416.078 5416.078 5416.078 5416.078 5416.078
## 289 290 291 292 293 294 295 296
## 5416.078 5416.078 5416.078 5416.078 5416.078 5416.078 5416.078 5416.078
## 297 298 299 300 301 302 303 304
## 5416.078 5416.078 5416.078 5416.078 5416.078 5416.078 5416.078 5416.078
## 305
## 5416.078# Residuals
residuals(fit)## 1 2 3 4 5
## -1318.874809 6215.159113 -1398.773045 -1130.959612 1461.226955
## 6 7 8 9 10
## 1146.159113 -2186.773045 -1130.959612 3247.226955 1461.159113
## 11 12 13 14 15
## -2058.874809 2301.226955 -211.959612 4506.159113 -1398.773045
## 16 17 18 19 20
## 1189.226955 4506.159113 -373.874809 7069.226955 -211.959612
## 21 22 23 24 25
## 4506.159113 -2215.874809 4504.193034 7069.226955 -211.959612
## 26 27 28 29 30
## -1398.773045 3247.226955 -1318.874809 4504.193034 8381.226955
## 31 32 33 34 35
## -1130.959612 4504.193034 -1130.959612 -1318.874809 8381.226955
## 36 37 38 39 40
## -1319.857848 6213.193034 -1130.959612 -1132.925691 2.142152
## 41 42 43 44 45
## 3664.193034 1189.226955 -1132.925691 -1132.925691 -1319.857848
## 46 47 48 49 50
## 4504.193034 -1319.857848 -213.925691 8199.226955 -1319.857848
## 51 52 53 54 55
## -213.925691 -1319.857848 -1376.925691 -1320.840887 1189.226955
## 56 57 58 59 60
## -2759.925691 -1320.840887 -2759.925691 -1320.840887 -686.908730
## 61 62 63 64 65
## 308.125191 -1396.806966 -1128.993533 2303.193034 3668.125191
## 66 67 68 69 70
## -1396.806966 -1128.993533 1359.193034 -6.874809 48.091270
## 71 72 73 74 75
## 1359.193034 -209.993533 518.125191 -1396.806966 351.193034
## 76 77 78 79 80
## -6.874809 3540.091270 351.193034 -209.993533 -6.874809
## 81 82 83 84 85
## 48.091270 3666.159113 351.193034 -209.993533 -1396.806966
## 86 87 88 89 90
## 623.193034 -686.908730 306.159113 1191.193034 -1128.993533
## 91 92 93 94 95
## 306.159113 -1128.993533 -686.908730 351.193034 -687.891769
## 96 97 98 99 100
## 5375.159113 -1128.993533 -767.959612 2069.108231 -218.840887
## 101 102 103 104 105
## 351.193034 -1130.959612 -1130.959612 -687.891769 831.159113
## 106 107 108 109 110
## -687.891769 331.040388 -344.840887 351.193034 -687.891769
## 111 112 113 114 115
## -211.959612 -687.891769 -211.959612 -344.840887 -688.874809
## 116 117 118 119 120
## -803.806966 -767.959612 -688.874809 -1130.959612 -248.840887
## 121 122 123 124 125
## -688.874809 -2681.467754 -2362.467754 -2362.467754 -2362.467754
## 126 127 128 129 130
## -2362.467754 -2362.467754 -2681.467754 -2681.467754 -1840.467754
## 131 132 133 134 135
## -1840.467754 -1840.467754 -1840.467754 -1105.467754 -2260.467754
## 136 137 138 139 140
## -1105.467754 -580.467754 -895.467754 -580.467754 -580.467754
## 141 142 143 144 145
## -580.467754 -1789.467754 -2078.467754 -1422.467754 -855.467754
## 146 147 148 149 150
## -1422.467754 -1422.467754 -2260.467754 -360.467754 -2774.467754
## 151 152 153 154 155
## -865.467754 -865.467754 -865.467754 -1364.467754 -366.467754
## 156 157 158 159 160
## -366.467754 -1836.467754 -760.467754 -1836.467754 -1182.467754
## 161 162 163 164 165
## 2283.532246 -1312.467754 -823.467754 -823.467754 -823.467754
## 166 167 168 169 170
## -2204.467754 -1706.467754 -823.467754 -823.467754 -298.467754
## 171 172 173 174 175
## -298.467754 -298.467754 -2786.467754 -2412.467754 -1658.467754
## 176 177 178 179 180
## -2362.467754 -2362.467754 -2199.467754 -2412.467754 -2786.467754
## 181 182 183 184 185
## 3509.007144 10573.007144 12594.007144 10573.007144 10573.007144
## 186 187 188 189 190
## 2546.007144 1660.007144 -823.992856 -1186.992856 -823.992856
## 191 192 193 194 195
## -823.992856 -267.992856 782.007144 2966.007144 -267.992856
## 196 197 198 199 200
## -267.992856 -1967.992856 -2704.077659 -1681.077659 -1967.992856
## 201 202 203 204 205
## -1681.077659 -1967.992856 -2111.077659 -1169.992856 -1182.077659
## 206 207 208 209 210
## -1967.992856 -1681.077659 -1681.077659 -1867.992856 -819.077659
## 211 212 213 214 215
## 1901.007144 3513.007144 1901.007144 2066.007144 1649.007144
## 216 217 218 219 220
## 2426.007144 -792.992856 2588.007144 4142.007144 4142.007144
## 221 222 223 224 225
## 3512.007144 3512.007144 -1426.077659 -819.077659 -1426.077659
## 226 227 228 229 230
## -1681.077659 -2111.077659 -263.077659 -712.992856 -1681.077659
## 231 232 233 234 235
## 258.007144 -2222.077659 -1169.992856 204.007144 204.007144
## 236 237 238 239 240
## 204.007144 -792.992856 204.007144 2063.007144 2777.007144
## 241 242 243 244 245
## 2063.007144 204.007144 -1165.077659 -1165.077659 -1165.077659
## 246 247 248 249 250
## -1165.077659 262.922341 -708.077659 -183.992856 -683.992856
## 251 252 253 254 255
## 1286.007144 3306.007144 -683.992856 2230.007144 -588.992856
## 256 257 258 259 260
## -588.992856 -588.992856 1286.007144 682.007144 724.007144
## 261 262 263 264 265
## -63.992856 -708.077659 -393.077659 146.007144 -708.077659
## 266 267 268 269 270
## 1653.922341 2036.007144 1653.922341 -393.077659 724.007144
## 271 272 273 274 275
## -63.992856 -788.077659 -788.077659 -998.077659 -788.077659
## 276 277 278 279 280
## 208.922341 -473.077659 -1165.077659 -708.077659 -1863.077659
## 281 282 283 284 285
## -106.077659 208.922341 -788.077659 -788.077659 -1418.077659
## 286 287 288 289 290
## -998.077659 -998.077659 -1786.077659 -1418.077659 -2173.077659
## 291 292 293 294 295
## -1623.077659 -1333.077659 -2043.077659 -1203.077659 -1753.077659
## 296 297 298 299 300
## 728.922341 -584.077659 -2173.077659 -2359.077659 -584.077659
## 301 302 303 304 305
## -59.077659 -584.077659 -584.077659 -584.077659 -1161.077659# Statistical Significance and p-values
summary(fit)##
## Call:
## lm(formula = yPrice ~ xAdvBookDays, data = price1)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2786.5 -1320.8 -688.9 351.2 12594.0
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 5422.959 224.497 24.16 <2e-16 ***
## xAdvBookDays -0.983 6.154 -0.16 0.873
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2392 on 303 degrees of freedom
## Multiple R-squared: 8.422e-05, Adjusted R-squared: -0.003216
## F-statistic: 0.02552 on 1 and 303 DF, p-value: 0.8732