Ram_Final
2023-03-02
- A single person is randomly selected for jury duty. What is the probability that this person will have an IQ of 110 or higher? Be sure to write the probability statement and show your R code.
x = pnorm(110, mean=100, sd=16,lower.tail = FALSE)
mean(x)## [1] 0.2659855- lower.tail = FALSE indicates we are looking at the right side tail.
library(ggplot2)
ggplot(data.frame(x),aes(x)) + geom_histogram() ## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
n<-110
mean<-100
sigma=16
x=pnorm(110,mean=100,sd=16,lower.tail = FALSE)
hist(x)
mean(x)## [1] 0.2659855If you draw samples from a normal distribution, then the distribution of sample means is also normal. The mean of the distribution of sample means is identical to the mean of the “parent population,” the population from which the samples are drawn. The higher the sample size that is drawn, the “narrower” will be the spread of the distribution of sample means.
2.
Is college worth it? Among a simple random sample of 331 American adults who do not have a four-year college degree and are not currently enrolled in school, 48% said they decided not to go to college because they could not afford school.
- A newspaper article states that only a minority of the Americans who decide not to go to college do so because they cannot afford it and uses the point estimate from this survey as evidence. Conduct a hypothesis test with α=.05 to determine if these data provide strong evidence that more than 50% adult Americans do not go to college for financial reasons. What is the critical value, test statistic, and p-value. Do you reject or fail to reject the null ? Step 1: Null/Alternative hypothesis - Ho: p = .5 Ha: p ≠ .5 Step 2: Significance level Step 3: Distribution Step 4: Test Statistic from sample
- Would you expect a confidence interval (α=.05 , double sided) for the proportion of American adults who decide not to go to college for financial reasons to include 0.5 (hypothesized value)? Show, or explain.
Ho: p = .5 Ha: p ≠ .5 Ho: p = .5 Ha: p < .5 Ho: p = .5 Ha: p > .5
Ho: >= 48% of American adults who decided not to go to college did so because they could not afford it
Ha: < 48 of American adults who decided not to go to college did so because they could not afford it
t is the test statistic pie is the population proportion = 0.50 p is the sample proportion = 159 / 331 = 0.48 n is the sample size = 331
Decide Alpha
alpha<-0.05
Calculating the Se Standard Error
Se<-sqrt(sd/sqrt(n) se
phat = .48
p01 = .50
n = 331
# the test statistic is a z=(p hat-p)/sqrt(p(1-p)/n)
zee1 <-(phat-p01)/sqrt(p01*(1-p01)/n)
zee1## [1] -0.7277362phat = .48
p01 = .50
n = 331
Se<-sqrt((phat*(1-phat))/n)
Se## [1] 0.02746049With a standard error of .027 or 2.7% and a sample proportion of .48 or 48% we can form a 95% confidence interval. We are 95% confident that the population proportion falls within 2 SE’s of the sample proportion, in the range (42.6%-53.4%). Therefore the confidence interval, at 95%, does contain .5.
phat = .48
p01 = .50
n = 331
# the test statistic is a z=(p hat-p)/sqrt(p(1-p)/n)
zee1 <-(phat-p01)/sqrt(p01*(1-p01)/n)
zee1## [1] -0.7277362pval<-pnorm(zee1)
pval## [1] 0.2333875pvalue<-2*(1-pval)
pvalue## [1] 1.533225alpha<-0.05
alpha## [1] 0.05# compute critical value (split alpha since we have double sided hypothesis)
?pnorm## starting httpd help server ... donecritical_value <- qnorm(p = .975,
mean = 0,
sd = 1
)
critical_value## [1] 1.959964#find Z critical value. Another way to find critical value
Zcriric<-qnorm(p=.05/2, lower.tail=FALSE)
Zcriric## [1] 1.959964##The smaller the p-value, the stronger the evidence that you should reject the null hypothesis.
## A p-value less than 0.05 (typically ≤ 0.05) is statistically significant.`myvector=c(4,30)
mymatrix=matrix(c(4,30,24,45), nrow=2)
colnames(mymatrix) <- c("Control", "Treatment")
rownames(mymatrix) <-c("Alive", "Dead")
mymatrix## Control Treatment
## Alive 4 24
## Dead 30 45- #Question 3: Hypothesis testing- Difference of two means Gaming and distracted eating. You are investigating the effects of being distracted by a game on how much people eat. The 22 patients in the treatment group who ate their lunch while playing solitaire were asked to do a serial-order recall of the food lunch items they ate. The average number of items recalled by the patients in this group was 4.9, with a standard deviation of 1.8. The average number of items recalled by the patients in the control group (no distraction, n=22 too) was 6.1, with a standard deviation of 1.8.
Do these data provide strong evidence that the average number of food items recalled by the patients in the treatment and control groups are different? Assume α=5%.
n1<-22
sd1<-1.8
mean.non<-4.9
sd2<-1.8
n2<-22
n.non<-6.1
meandiff<-n1 - mean.non
SE<-sqrt((sd1^2/n2)+(sd2^2/n.non))
df<-n.non-1
Tstat<-meandiff/SE
tdf<-qt(p=.05, df, lower.tail=FALSE)
pvalue<-2*pt(Tstat, df, lower.tail = FALSE)
pvalue## [1] 4.000813e-06#Print the appropriate answer based on the p-value
4.000813e-06## [1] 4.000813e-06Because 4.000 below the range we can reject the null hypothesis
5Question 5: Assumptions Heart transplant success. The Stanford University Heart Transplant Study was conducted to determine whether an experimental heart transplant program increased lifespan. Each patient entering the program was officially designated a heart transplant candidate, meaning that he was gravely ill and might benefit from a new heart. Patients were randomly assigned into treatment and control groups. Patients in the treatment group received a transplant, and those in the control group did not. The table below displays how many patients survived and died in each group -
mym <- matrix(c(4,30,24,45), nrow=2) colnames(mym) <- c(“control”, “treatment”) rownames(mym) <- c(“alive”,“dead”) mym ## control treatment ## alive 4 24 ## dead 30 45 Suppose we are interested in estimating the difference in survival rate between the control and treatment groups using a confidence interval. Explain why we cannot construct such an interval using the normal approximation. What might go wrong if we constructed the confidence interval despite this problem?
alive = c(4,24)
dead = c(30,45)
Data1<-rbind(alive,dead)
Data1## [,1] [,2]
## alive 4 24
## dead 30 45colnames(Data1)=c("control","treatment")
chisq.test(Data1)##
## Pearson's Chi-squared test with Yates' continuity correction
##
## data: Data1
## X-squared = 4.9891, df = 1, p-value = 0.02551chisq.test(cbind(alive,dead))##
## Pearson's Chi-squared test with Yates' continuity correction
##
## data: cbind(alive, dead)
## X-squared = 4.9891, df = 1, p-value = 0.02551#Rejectv H0 Null Hypothesis #4,Question 4: Working backwards. A 90% confidence interval for a population mean is (65,77). The population distribution is approximately normal and the population standard deviation is unknown. This confidence interval is based on a simple random sample of 25 observations (double sided). Calculate the sample mean, the margin of error, and the sample standard deviation.
SAMPLE MEAN
sample mean = X2+X12
MARGING OF ERROR
Marging of Error = X2−X12
n<- 25
phat <- (65+77)/2
phat## [1] 71Mean <-(77-65)/2
Mean## [1] 6df <- 25-1
t.value <- qt(.95, df)
t.value## [1] 1.710882tdf <- round(qt(c(.05, .95), df=24)[2], 3)
tdf## [1] 1.711Se <- round((77-phat)/tdf, 3)
Se## [1] 3.507sd <- Se * sqrt(25)
sd## [1] 17.535sd1 <- (Mean/t.value)*5
sd1## [1] 17.53481- Question 6: Data Question The sinking of the Titanic is one of the most infamous shipwrecks in history.
On April 15, 1912, during her maiden voyage, the widely considered “unsinkable” RMS Titanic sank after colliding with an iceberg. Unfortunately, there weren’t enough lifeboats for everyone onboard, resulting in the death of 1502 out of 2224 passengers and crew.
While there was some element of luck involved in surviving, it seems some groups of people were more likely to survive than others.
In this question, you are to build a predictive model that answers the question: “what sorts of people were more likely to survive?” using passenger data (ie name, age, gender, socio-economic class, etc).
Import the titanic train.csv file in
library(ggfortify)
library(ggplot2)
library(readr)
library(dplyr)##
## Attaching package: 'dplyr'## The following objects are masked from 'package:stats':
##
## filter, lag## The following objects are masked from 'package:base':
##
## intersect, setdiff, setequal, union library(tidyr)
library(viridis)## Loading required package: viridisLite library(ggthemes)
library(ggalt)## Registered S3 methods overwritten by 'ggalt':
## method from
## fortify.table ggfortify
## grid.draw.absoluteGrob ggplot2
## grobHeight.absoluteGrob ggplot2
## grobWidth.absoluteGrob ggplot2
## grobX.absoluteGrob ggplot2
## grobY.absoluteGrob ggplot2traind <- read.table(file="C:/R/final/trainfinal.csv", header=TRUE, sep=",", stringsAsFactors = TRUE)
str(traind)## 'data.frame': 891 obs. of 12 variables:
## $ PassengerId: int 1 2 3 4 5 6 7 8 9 10 ...
## $ Survived : int 0 1 1 1 0 0 0 0 1 1 ...
## $ Pclass : int 3 1 3 1 3 3 1 3 3 2 ...
## $ Name : Factor w/ 891 levels "Abbing, Mr. Anthony",..: 109 191 358 277 16 559 520 629 417 581 ...
## $ Sex : Factor w/ 2 levels "female","male": 2 1 1 1 2 2 2 2 1 1 ...
## $ Age : num 22 38 26 35 35 NA 54 2 27 14 ...
## $ SibSp : int 1 1 0 1 0 0 0 3 0 1 ...
## $ Parch : int 0 0 0 0 0 0 0 1 2 0 ...
## $ Ticket : Factor w/ 681 levels "110152","110413",..: 524 597 670 50 473 276 86 396 345 133 ...
## $ Fare : num 7.25 71.28 7.92 53.1 8.05 ...
## $ Cabin : Factor w/ 148 levels "","A10","A14",..: 1 83 1 57 1 1 131 1 1 1 ...
## $ Embarked : Factor w/ 4 levels "","C","Q","S": 4 2 4 4 4 3 4 4 4 2 ...set.seed(100)
#loading psych package
require(psych)## Loading required package: psych##
## Attaching package: 'psych'## The following objects are masked from 'package:ggplot2':
##
## %+%, alphatrainc<- na.omit(traind)
Traindesc <- describe(trainc) # Summary Statistics
Traindesc ## vars n mean sd median trimmed mad min max range
## PassengerId 1 714 448.58 259.12 445.00 448.76 337.29 1.00 891.00 890.00
## Survived 2 714 0.41 0.49 0.00 0.38 0.00 0.00 1.00 1.00
## Pclass 3 714 2.24 0.84 2.00 2.30 1.48 1.00 3.00 2.00
## Name* 4 714 422.68 263.59 397.50 418.23 341.00 1.00 891.00 890.00
## Sex* 5 714 1.63 0.48 2.00 1.67 0.00 1.00 2.00 1.00
## Age 6 714 29.70 14.53 28.00 29.27 13.34 0.42 80.00 79.58
## SibSp 7 714 0.51 0.93 0.00 0.30 0.00 0.00 5.00 5.00
## Parch 8 714 0.43 0.85 0.00 0.23 0.00 0.00 6.00 6.00
## Ticket* 9 714 336.39 203.43 332.00 335.62 278.73 1.00 681.00 680.00
## Fare 10 714 34.69 52.92 15.74 23.19 12.21 0.00 512.33 512.33
## Cabin* 11 714 21.11 40.20 1.00 10.89 0.00 1.00 148.00 147.00
## Embarked* 12 714 3.59 0.79 4.00 3.74 0.00 1.00 4.00 3.00
## skew kurtosis se
## PassengerId 0.00 -1.23 9.70
## Survived 0.38 -1.86 0.02
## Pclass -0.47 -1.42 0.03
## Name* 0.13 -1.24 9.86
## Sex* -0.56 -1.69 0.02
## Age 0.39 0.16 0.54
## SibSp 2.51 6.96 0.03
## Parch 2.61 8.75 0.03
## Ticket* 0.05 -1.30 7.61
## Fare 4.63 30.61 1.98
## Cabin* 1.88 2.17 1.50
## Embarked* -1.48 0.37 0.03ggplot(traind, aes(Pclass)) + geom_density(fill="blue")
ggplot(traind, aes(log(Pclass))) + geom_density(fill="blue")
ggplot(traind, aes(sqrt(Pclass))) + geom_density(fill="blue")
slope <-cor(trainc$Pclass,trainc$Fare) * (sd(trainc$Pclass)/sd(trainc$Fare))
slope## [1] -0.008778397intercept <- mean(trainc$Pclass) - (slope * mean(trainc$Fare))
intercept## [1] 2.541257library('tidyverse')## ── Attaching packages ─────────────────────────────────────── tidyverse 1.3.2 ──
## ✔ tibble 3.1.8 ✔ stringr 1.5.0
## ✔ purrr 1.0.1 ✔ forcats 1.0.0
## ── Conflicts ────────────────────────────────────────── tidyverse_conflicts() ──
## ✖ psych::%+%() masks ggplot2::%+%()
## ✖ psych::alpha() masks ggplot2::alpha()
## ✖ dplyr::filter() masks stats::filter()
## ✖ dplyr::lag() masks stats::lag()library('ggplot2')
library('dplyr')
trainc %>%
ggplot(aes(x = trainc$Fare, y = trainc$Pclass)) +
geom_point(colour = "red")
trainc[is.na(trainc)] <- 0
plot(trainc)
##cor(trainc, use="pairwise.complete.obs")cor(trainc$Pclass,trainc$Fare)## [1] -0.5541825trainc %>%
ggplot(aes(x = sqrt(Pclass), y = sqrt(Fare))) +
geom_point(colour = "orangered") 
trainc %>% ggplot(aes(x = sqrt(Fare), y = sqrt(Pclass))) +
geom_point(colour = “maroon”) + geom_smooth(method = “lm”, fill =
NA)
```r
hist(x = trainc$Fare, xlab = "", main = "Outpatients (RVU)")
hist(x=trainc$Pclass,xlab = "",main = "trainc Pclass")
plot(x = trainc$Fare, y = trainc$Pclass, xlab = "Fare", ylab = "Pclass")
univariate_reg = lm(trainc$Pclass ~ trainc$Fare)
summary(univariate_reg)##
## Call:
## lm(formula = trainc$Pclass ~ trainc$Fare)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.5413 -0.4403 0.5158 0.5301 2.9562
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 2.5412569 0.0312531 81.31 <2e-16 ***
## trainc$Fare -0.0087784 0.0004941 -17.77 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.6982 on 712 degrees of freedom
## Multiple R-squared: 0.3071, Adjusted R-squared: 0.3061
## F-statistic: 315.6 on 1 and 712 DF, p-value: < 2.2e-16options(scipen = 999)
summary(univariate_reg)##
## Call:
## lm(formula = trainc$Pclass ~ trainc$Fare)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.5413 -0.4403 0.5158 0.5301 2.9562
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 2.5412569 0.0312531 81.31 <0.0000000000000002 ***
## trainc$Fare -0.0087784 0.0004941 -17.77 <0.0000000000000002 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.6982 on 712 degrees of freedom
## Multiple R-squared: 0.3071, Adjusted R-squared: 0.3061
## F-statistic: 315.6 on 1 and 712 DF, p-value: < 0.00000000000000022?abline
abline(reg = univariate_reg, col="blue")
model<-lm(Pclass ~ Fare,data=trainc)
summary(model)##
## Call:
## lm(formula = Pclass ~ Fare, data = trainc)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.5413 -0.4403 0.5158 0.5301 2.9562
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 2.5412569 0.0312531 81.31 <0.0000000000000002 ***
## Fare -0.0087784 0.0004941 -17.77 <0.0000000000000002 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.6982 on 712 degrees of freedom
## Multiple R-squared: 0.3071, Adjusted R-squared: 0.3061
## F-statistic: 315.6 on 1 and 712 DF, p-value: < 0.00000000000000022lmodel <- lm(sqrt(Fare) ~ sqrt(Pclass), data = trainc)
lmodel##
## Call:
## lm(formula = sqrt(Fare) ~ sqrt(Pclass), data = trainc)
##
## Coefficients:
## (Intercept) sqrt(Pclass)
## 15.247 -6.963lmodel$coefficients## (Intercept) sqrt(Pclass)
## 15.246911 -6.962825summary(lmodel)##
## Call:
## lm(formula = sqrt(Fare) ~ sqrt(Pclass), data = trainc)
##
## Residuals:
## Min 1Q Median 3Q Max
## -8.2841 -0.8174 -0.3497 0.6622 14.3506
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 15.2469 0.3998 38.14 <0.0000000000000002 ***
## sqrt(Pclass) -6.9628 0.2673 -26.05 <0.0000000000000002 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.171 on 712 degrees of freedom
## Multiple R-squared: 0.4879, Adjusted R-squared: 0.4872
## F-statistic: 678.4 on 1 and 712 DF, p-value: < 0.00000000000000022Fvalue<-fitted.values(model)
Fvalue## 1 2 3 4 5 7 8
## 2.4776135 1.9155038 2.4716881 2.0751240 2.4705908 2.0859873 2.3562522
## 9 10 11 12 13 14 15
## 2.4435244 2.2772835 2.3946577 2.3081905 2.4705908 2.2667125 2.4723096
## 16 17 19 21 22 23 24
## 2.4008025 2.2855861 2.3832458 2.3130186 2.4271377 2.4707734 2.2296238
## 25 26 28 31 34 35 36
## 2.3562522 2.2657250 0.2325384 2.2979127 2.4490837 1.8199290 2.0847802
## 38 39 40 41 42 44 45
## 2.4705908 2.3832458 2.4425728 2.4580816 2.3569106 2.1762582 2.4720902
## 50 51 52 53 54 55 57
## 2.3850014 2.1928643 2.4727854 1.8676975 2.3130186 1.9971789 2.4490837
## 58 59 60 61 62 63 64
## 2.4777961 2.2976564 2.1295501 2.4777961 1.8389851 1.8084802 2.2963396
## 67 68 69 70 71 72 73
## 2.4490837 2.4696401 2.4716881 2.4652140 2.4490837 2.1295501 1.8960447
## 74 75 76 79 80 81 82
## 2.4143722 2.0453143 2.4741022 2.2866834 2.4317464 2.4622513 2.4578621
## 84 85 86 87 89 90 91
## 2.1277944 2.4490837 2.4021193 2.2394995 0.2325384 2.4705908 2.4705908
## 92 93 94 95 97 98 99
## 2.4723096 2.0042385 2.3606414 2.4776135 2.2370486 1.9850726 2.3393538
## 100 101 103 104 105 106 107
## 2.3130186 2.4719444 1.8627965 2.4652869 2.4716881 2.4719444 2.4741022
## 109 111 112 113 114 115 116
## 2.4719444 2.0847802 2.4143722 2.4705908 2.4550091 2.4143362 2.4716881
## 117 118 119 120 121 123 124
## 2.4732243 2.3569106 0.3684210 2.2667125 1.8960447 2.2772835 2.4271377
## 125 126 128 130 131 132 133
## 1.8627965 2.4425728 2.4785642 2.4800276 2.4719444 2.4793692 2.4139701
## 134 135 136 137 138 139 140
## 2.3130186 2.4271377 2.4091789 2.3105317 2.0751240 2.4603490 1.8460078
## 142 143 144 145 146 147 148
## 2.4732243 2.4021193 2.4820027 2.4403053 2.2186508 2.4728223 2.2394995
## 149 150 151 152 153 154 156
## 2.3130186 2.4271377 2.4313075 1.9566157 2.4705908 2.4139701 2.0024459
## 157 158 161 162 163 164 165
## 2.4733709 2.4705908 2.3999247 2.4029971 2.4730049 2.4652140 2.1928643
## 166 168 170 171 172 173 174
## 2.3610803 2.2963396 2.0453143 2.2471806 2.2855861 2.4435244 2.4716881
## 175 176 178 179 180 183 184
## 2.2717970 2.4723096 2.2892072 2.4271377 2.5412569 2.2657250 2.1988994
## 185 188 189 190 191 192 193
## 2.3479127 2.3081905 2.4051917 2.4719444 2.4271377 2.4271377 2.4723096
## 194 195 196 198 200 201 203
## 2.3130186 2.2979127 1.2550391 2.4674815 2.4271377 2.4578621 2.4842342
## 204 205 206 207 208 209 210
## 2.4778330 2.4705908 2.4494129 2.4021193 2.3763328 2.4732243 2.2691266
## 211 212 213 214 216 217 218
## 2.4793692 2.3569106 2.4776135 2.4271377 1.5468840 2.4716881 2.3042402
## 219 220 221 222 223 225 226
## 1.8715381 2.4490837 2.4705908 2.4271377 2.4705908 1.7512012 2.4591789
## 227 228 229 231 232 233 234
## 2.4490837 2.4776135 2.4271377 1.8084802 2.4730049 2.4227485 2.2657250
## 235 237 238 239 240 243 244
## 2.4490837 2.3130186 2.3108240 2.4490837 2.4335021 2.4490837 2.4787108
## 245 246 247 248 249 250 252
## 2.4778330 1.7512012 2.4730049 2.4139701 2.0799153 2.3130186 2.4494129
## 253 254 255 256 258 259 260
## 2.3081905 2.3999247 2.3638235 2.4074232 1.7819255 -1.9561723 2.3130186
## 262 263 264 266 267 268 269
## 2.2657250 1.8420576 2.5412569 2.4490837 2.1928643 2.4730049 1.1941021
## 270 272 273 274 276 277 279
## 1.3506139 2.5412569 2.3700782 2.2805385 1.8569080 2.4732243 2.2855861
## 280 281 282 283 284 286 287
## 2.3634944 2.4732243 2.4723096 2.4578621 2.4705908 2.4652140 2.4578621
## 288 289 290 291 292 293 294
## 2.4719444 2.4271377 2.4732243 1.8490803 1.7417275 2.4282350 2.4635681
## 295 297 298 300 303 306 308
## 2.4719444 2.4777961 1.2108908 0.3684210 2.5412569 1.2108908 1.5852895
## 309 310 311 312 313 314 315
## 2.3305754 2.0415098 1.8112603 0.2380249 2.3130186 2.4719444 2.3108240
## 316 317 318 319 320 321 322
## 2.4723096 2.3130186 2.4183593 1.0939915 1.3605625 2.4776135 2.4719444
## 323 324 326 327 328 329 330
## 2.4328437 2.2866834 1.3506139 2.4865016 2.4271377 2.3610803 2.0322925
## 332 333 334 337 338 339 340
## 2.2910726 1.1941021 2.3832458 1.9566157 1.3605625 2.4705908 2.2296238
## 341 342 343 344 345 346 347
## 2.3130186 0.2325384 2.4271377 2.4271377 2.4271377 2.4271377 2.4271377
## 349 350 351 353 354 356 357
## 2.4016804 2.4652140 2.4602762 2.4777961 2.3850014 2.4578621 2.0584451
## 358 361 362 363 364 366 367
## 2.4271377 2.2963396 2.2979127 2.4143722 2.4793692 2.4776135 1.8806825
## 370 371 372 373 374 375 377
## 1.9329140 2.0545676 2.4842342 2.4705908 1.3506139 2.3562522 2.4776135
## 378 379 380 381 382 383 384
## 0.6846259 2.5060336 2.4730049 0.5439521 2.4030700 2.4716881 2.0847802
## 386 387 388 390 391 392 393
## 1.8960447 2.1295501 2.4271377 2.4359161 1.4878492 2.4728223 2.4716881
## 394 395 396 397 398 399 400
## 1.5468840 2.3946577 2.4728223 2.4723096 2.3130186 2.4490837 2.4302102
## 401 402 403 404 405 406 407
## 2.4716881 2.4705908 2.4550091 2.4021193 2.4652140 2.3569106 2.4732243
## 408 409 413 415 417 418 419
## 2.3766620 2.4730049 1.7512012 2.4716881 2.2559590 2.4271377 2.4271377
## 420 422 423 424 425 427 428
## 2.3292586 2.4733709 2.4721270 2.4148480 2.3638235 2.3130186 2.3130186
## 430 431 433 434 435 436 437
## 2.4705908 2.3081905 2.3130186 2.4787108 2.0505445 1.4878492 2.2394995
## 438 439 440 441 442 443 444
## 2.3766620 0.2325384 2.4490837 2.3108240 2.4578621 2.4730049 2.4271377
## 446 447 448 449 450 451 453
## 1.8226722 2.3700782 2.3081905 2.3721999 2.2735158 2.2976564 2.2976564
## 454 456 457 459 461 462 463
## 1.7590648 2.4719444 2.3081905 2.4490837 2.3081905 2.4705908 2.2032886
## 464 466 468 470 472 473 474
## 2.4271377 2.4793692 2.3081905 2.3721999 2.4652140 2.2976564 2.4201879
## 475 477 478 479 480 481 483
## 2.4548994 2.3569106 2.4794061 2.4752363 2.4333923 2.1295501 2.4705908
## 484 485 487 488 489 490 492
## 2.4570940 1.7417275 1.7512012 2.2805385 2.4705908 2.4016804 2.4776135
## 493 494 495 497 499 500 501
## 2.2735158 2.1066894 2.4705908 1.8542007 1.2108908 2.4728223 2.4652140
## 502 504 505 506 507 509 510
## 2.4732243 2.4570940 1.7819255 1.5852895 2.3130186 2.3435235 2.0453143
## 511 513 514 515 516 517 519
## 2.4732243 2.3104948 2.0198201 2.4754558 2.2426088 2.4490837 2.3130186
## 520 521 522 524 526 527 529
## 2.4719444 1.7204768 2.4719444 2.0322925 2.4732243 2.4490837 2.4716881
## 530 531 533 535 536 537 538
## 2.4403053 2.3130186 2.4777961 2.4652140 2.3108240 2.3081905 1.6070160
## 540 541 542 543 544 545 546
## 2.1067262 1.9179907 2.2667125 2.2667125 2.3130186 1.6070160 2.3130186
## 547 549 550 551 552 554 555
## 2.3130186 2.3610803 2.2186508 1.5678793 2.3130186 2.4778330 2.4730049
## 556 557 559 560 562 563 566
## 2.3081905 2.1936324 1.8420576 2.3885128 2.4719444 2.4227485 2.3292586
## 567 568 570 571 572 573 575
## 2.4719444 2.3562522 2.4723096 2.4490837 2.0893520 2.3096169 2.4705908
## 576 577 578 580 581 582 583
## 2.4139701 2.4271377 2.0505445 2.4716881 2.2779050 1.5678793 2.3130186
## 584 586 587 588 589 591 592
## 2.1890237 1.8420576 2.4095809 1.8460078 2.4705908 2.4787108 1.8542007
## 593 595 596 598 600 601 604
## 2.4776135 2.3130186 2.3292586 2.5412569 2.0415098 2.3042402 2.4705908
## 605 606 607 608 609 610 611
## 2.3081905 2.4047528 2.4719444 2.2735158 2.1762582 1.1941021 2.2667125
## 615 616 617 618 619 620 621
## 2.4705908 1.9706611 2.4148480 2.3999247 2.1988994 2.4490837 2.4143722
## 622 623 624 625 626 627 628
## 2.0799153 2.4030700 2.4723096 2.3999247 2.2575321 2.4328437 1.8569080
## 629 631 632 633 635 636 637
## 2.4719444 2.2779050 2.4793323 2.2735158 2.2963396 2.4271377 2.4716881
## 638 639 641 642 643 645 646
## 2.3108240 2.1928643 2.4723096 1.9329140 2.2963396 2.3721999 1.8676975
## 647 648 650 652 653 655 656
## 2.4719444 2.2296238 2.4749800 2.3393538 2.4672260 2.4820027 1.8960447
## 658 659 660 661 662 663 664
## 2.4051917 2.4271377 1.5468840 1.3680241 2.4778330 2.3166397 2.4754558
## 665 666 667 669 671 672 673
## 2.4716881 1.8960447 2.4271377 2.4705908 2.1988994 2.0847802 2.4490837
## 674 676 677 678 679 680 682
## 2.4271377 2.4730049 2.4705908 2.4548625 2.1295501 -1.9561723 1.8676975
## 683 684 685 686 687 688 689
## 2.4602762 2.1295501 2.1988994 2.1762582 2.1928643 2.4519736 2.4728223
## 690 691 692 694 695 696 697
## 0.6860524 2.0408883 2.4234798 2.4778330 2.3081905 2.4227485 2.4705908
## 699 700 701 702 703 704 705
## 1.5678793 2.4741022 0.5439521 2.3104948 2.4143722 2.4732972 2.4723096
## 706 707 708 709 711 713 714
## 2.3130186 2.4227485 2.3104948 1.2108908 2.1066894 2.0847802 2.4580087
## 715 716 717 718 720 721 722
## 2.4271377 2.4741022 0.5439521 2.4490837 2.4730049 2.2515698 2.4793323
## 723 724 725 726 727 729 730
## 2.4271377 2.4271377 2.0751240 2.4652140 2.3569106 2.3130186 2.4716881
## 731 732 734 735 736 737 738
## 0.6860524 2.3763328 2.4271377 2.4271377 2.3999247 2.2394995 -1.9561723
## 742 743 744 745 746 747 748
## 1.8490803 0.2380249 2.3999247 2.4716881 1.9179907 2.3634944 2.4271377
## 749 750 751 752 753 754 755
## 2.0751240 2.4732243 2.3393538 2.4317464 2.4578621 2.4719444 1.9706611
## 756 757 758 759 760 762 763
## 2.4139701 2.4728223 2.4403053 2.4705908 1.7819255 2.4787108 2.4777961
## 764 765 766 768 770 771 772
## 1.4878492 2.4730049 1.8569080 2.4732243 2.4678476 2.4578621 2.4723096
## 773 775 776 778 780 781 782
## 2.4490837 2.3393538 2.4732243 2.4317464 0.6860524 2.4777961 2.0408883
## 783 785 786 787 788 789 790
## 2.2779050 2.4793692 2.4776135 2.4754558 2.2855861 2.3606414 1.8460078
## 792 795 796 797 798 799 800
## 2.3130186 2.4719444 2.4271377 2.3136401 2.4650314 2.4777961 2.3292586
## 801 802 803 804 805 806 807
## 2.4271377 2.3108240 1.4878492 2.4664939 2.4800276 2.4730049 2.5412569
## 808 809 810 811 812 813 814
## 2.4730049 2.4271377 2.0751240 2.4720173 2.3292586 2.4490837 2.2667125
## 815 817 818 819 820 821 822
## 2.4705908 2.4716881 2.2164193 2.4846362 2.2963396 1.7204768 2.4652140
## 823 824 825 828 830 831 832
## 2.5412569 2.4317464 2.1928643 2.2164193 1.8389851 2.4143722 2.3766620
## 834 835 836 837 839 841 842
## 2.4723096 2.4683962 1.8112603 2.4652140 2.0453143 2.4716881 2.4490837
## 843 844 845 846 848 849 851
## 2.2691266 2.4847460 2.4652140 2.4749800 2.4719444 2.2515698 2.2667125
## 852 853 854 855 856 857 858
## 2.4730049 2.4074232 2.1953881 2.3130186 2.4591789 1.0939915 2.3081905
## 859 861 862 863 865 866 867
## 2.3721999 2.4174086 2.4403053 2.3136401 2.4271377 2.4271377 2.4196032
## 868 870 871 872 873 874 875
## 2.0979847 2.4435244 2.4719444 2.0799153 2.4973649 2.4622513 2.3305754
## 876 877 878 880 881 882 883
## 2.4778330 2.4548266 2.4719444 1.8112603 2.3130186 2.4719444 2.4489371
## 884 885 886 887 888 890 891
## 2.4490837 2.4793692 2.2855861 2.4271377 2.2779050 2.2779050 2.4732243Res<-lm(formula = model, data = trainc)
Res##
## Call:
## lm(formula = model, data = trainc)
##
## Coefficients:
## (Intercept) Fare
## 2.541257 -0.008778# Residual Analysis
# plot(fitted(model),resid(model))
## plot(fitted(model),Res)
# abline(0,0)
qqnorm(resid(model))
qqline(resid(model))
par(mfrow=c(2,2))
plot(model)
Interpret the linear model - Pclass~Fare.
1 unit increase in Fare is associated with 235.1 units increase in Expenditures.
plot(x = univariate_reg)
qqnorm(y=trainc$Pclass)
qqnorm( y = log(trainc$Pclass))
Transformed regression - ln(Pclass)~RVU
hist(x = log(trainc$Pclass),xlab = "", main = "Log Pclass Details" )
hist(x=log(trainc$Fare), xlab = "", main = "Log Fare (Fare)")
plot(x = trainc$Fare, y = log (trainc$Pclass) , xlab = "Fare", ylab = "Log of Pclass")
univariate_reg_transformedY = lm( formula = log(trainc$Pclass) ~ trainc$Fare)
?abline
abline(reg = univariate_reg_transformedY, col="blue")
summary(univariate_reg)##
## Call:
## lm(formula = trainc$Pclass ~ trainc$Fare)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.5413 -0.4403 0.5158 0.5301 2.9562
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 2.5412569 0.0312531 81.31 <0.0000000000000002 ***
## trainc$Fare -0.0087784 0.0004941 -17.77 <0.0000000000000002 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.6982 on 712 degrees of freedom
## Multiple R-squared: 0.3071, Adjusted R-squared: 0.3061
## F-statistic: 315.6 on 1 and 712 DF, p-value: < 0.00000000000000022summary(univariate_reg_transformedY)##
## Call:
## lm(formula = log(trainc$Pclass) ~ trainc$Fare)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.8872 -0.1417 0.2456 0.2520 1.6677
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.8871916 0.0165851 53.49 <0.0000000000000002 ***
## trainc$Fare -0.0049868 0.0002622 -19.02 <0.0000000000000002 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.3705 on 712 degrees of freedom
## Multiple R-squared: 0.3368, Adjusted R-squared: 0.3359
## F-statistic: 361.7 on 1 and 712 DF, p-value: < 0.00000000000000022plot(univariate_reg_transformedY)
Constant variability The plot of residuals versus fitted observations
shows that the variability of errors around the predicted values is
slightly better.
Scale-Location plot - For OLS, the trend line is even and the residuals are uniformly scattered.
plot( univariate_reg , which = 3)
plot( univariate_reg_transformedY , which = 3)
hist(x = log(trainc$Pclass), xlab = "", main = "Log Pclass Details" )
hist(x = log(trainc$Fare) , xlab = "", main = "Log Pclass)")
#log_log_reg = lm(formula = log(trainc$Pclass) ~ log(trainc$Fare))
#summary(log_log_reg)# plot(log_log_reg) plot(x = trainc$Fare, y = log (trainc$Pclass) , xlab = "Fare", ylab = "Log of Pclass") 
trainc$Fare2 <- trainc$Fare^2
univariate_reg_transformedY2 = lm( formula = log(trainc$Pclass) ~ trainc$Fare+ trainc$Fare2)
# abline(reg = univariate_reg_transformedY2, col="blue")
# summary(univariate_reg)
# summary(univariate_reg_transformedY)
summary(univariate_reg_transformedY2)##
## Call:
## lm(formula = log(trainc$Pclass) ~ trainc$Fare + trainc$Fare2)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.0298 -0.1910 0.1506 0.1700 0.6512
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.029797839 0.016695350 61.68 <0.0000000000000002 ***
## trainc$Fare -0.011470684 0.000458865 -25.00 <0.0000000000000002 ***
## trainc$Fare2 0.000020586 0.000001271 16.20 <0.0000000000000002 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.3169 on 711 degrees of freedom
## Multiple R-squared: 0.5156, Adjusted R-squared: 0.5142
## F-statistic: 378.4 on 2 and 711 DF, p-value: < 0.00000000000000022plot(univariate_reg_transformedY2)
Some ways to fix heteroskedasticity - Transform the dependent variable. sqrt() will have larger penalty, but interpretation is not as easy/standard as when taking log.
Alternative models - sqrt(Pclass)~RVU
plot(x = trainc$Fare, y = sqrt(trainc$Pclass) , xlab = "Fare", ylab = "Square Root of Pclass")
univariate_reg_transformedY_sqrt = lm( formula = sqrt(trainc$Pclass) ~ trainc$Fare)
?abline
abline(reg = univariate_reg_transformedY_sqrt, col="blue")
summary(univariate_reg)##
## Call:
## lm(formula = trainc$Pclass ~ trainc$Fare)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.5413 -0.4403 0.5158 0.5301 2.9562
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 2.5412569 0.0312531 81.31 <0.0000000000000002 ***
## trainc$Fare -0.0087784 0.0004941 -17.77 <0.0000000000000002 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.6982 on 712 degrees of freedom
## Multiple R-squared: 0.3071, Adjusted R-squared: 0.3061
## F-statistic: 315.6 on 1 and 712 DF, p-value: < 0.00000000000000022summary(univariate_reg_transformedY)##
## Call:
## lm(formula = log(trainc$Pclass) ~ trainc$Fare)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.8872 -0.1417 0.2456 0.2520 1.6677
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.8871916 0.0165851 53.49 <0.0000000000000002 ***
## trainc$Fare -0.0049868 0.0002622 -19.02 <0.0000000000000002 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.3705 on 712 degrees of freedom
## Multiple R-squared: 0.3368, Adjusted R-squared: 0.3359
## F-statistic: 361.7 on 1 and 712 DF, p-value: < 0.00000000000000022summary(univariate_reg_transformedY_sqrt)##
## Call:
## lm(formula = sqrt(trainc$Pclass) ~ trainc$Fare)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.5777 -0.1292 0.1756 0.1809 1.0967
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.5777304 0.0112090 140.75 <0.0000000000000002 ***
## trainc$Fare -0.0032683 0.0001772 -18.44 <0.0000000000000002 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.2504 on 712 degrees of freedom
## Multiple R-squared: 0.3233, Adjusted R-squared: 0.3223
## F-statistic: 340.1 on 1 and 712 DF, p-value: < 0.00000000000000022plot( univariate_reg_transformedY_sqrt) 
```