Chapter Five Homework
Your Name Here
Tuesday, Nov 21, 2017 - 07:51:54 AM
Consider a population that has a normal distribution with mean \(\mu = 36\), standard deviation \(\sigma = 8\)
The sampling distribution of \(\bar{X}\) for samples of size 200 will have what distribution, mean, and standard error? (standard error of a statistic is the standard deviation of that statistic)
Use
Rto draw a random sample of size \(200\) from this population. Conduct EDA on your sample. (ch 5: shape of distribution [histogram, density plot] mean, standard deviation, shape, center, spread and bias, quantile for closeness to “normallity”)Compute the bootstrap distribution for your sample mean, and note the bootstrap mean and standard error.
Compare the bootstrap distribution to the theoretical sampling distribution by creating a table like Table 5.2.
Repeat parts a-d for sample sizes of \(n = 50\) and \(n = 10\). Carefully describe your observations about the effects of sample size on the bootstrap distribution.
Your answers:
The distribution is exactly normal. The mean is 36 and the standard error is 0.5656854
# Your code here
set.seed(13)
samp <- rnorm(200, 36, 8)
samp [1] 40.43462 33.75782 50.20131 37.49856 45.14021 39.32421 45.83605
[8] 37.89344 33.07694 44.84115 27.25125 39.69497 25.11212 21.15178
[15] 32.48116 34.44842 47.17145 36.80531 35.08449 41.61780 38.10034
[22] 50.68931 38.85922 27.63672 40.96147 37.19484 24.32547 19.78365
[29] 27.54434 30.17485 35.93431 42.78238 32.93207 31.78791 33.81419
[36] 31.15407 33.33706 34.06770 29.09780 29.22423 36.80272 48.72027
[43] 40.53196 48.91584 32.25080 30.19119 27.81329 20.49748 38.21718
[50] 47.26683 38.18503 42.04420 33.20785 31.63047 37.87490 33.61737
[57] 29.27619 42.61208 47.86953 41.59741 25.90741 38.38617 34.81754
[64] 28.88862 44.10453 28.63580 31.40884 45.20292 45.15060 34.08446
[71] 27.30558 35.50842 31.86642 20.73861 36.85725 26.58100 49.96342
[78] 32.81041 39.53952 39.60224 35.39150 38.38011 26.44516 20.02500
[85] 47.10810 35.34013 39.14012 27.33784 48.81696 44.03255 39.03917
[92] 31.47596 26.28978 25.08559 24.67094 33.95538 26.19659 37.71067
[99] 36.53779 42.85308 33.04662 40.89198 22.96994 37.13886 33.88246
[106] 31.98743 45.78724 39.45642 35.94552 49.38419 29.06787 49.08917
[113] 27.83118 29.50588 35.59390 35.09927 42.82120 41.97877 39.11575
[120] 34.51184 46.60385 39.63350 35.33321 35.42873 25.23603 34.39299
[127] 34.10929 35.53724 27.74448 29.06761 42.78089 36.98738 39.21218
[134] 43.37861 50.29275 34.56563 32.86013 41.77650 43.66819 25.63626
[141] 38.71489 26.81790 27.27266 20.05097 39.44207 49.65743 23.34199
[148] 33.14397 22.41077 26.78424 13.24189 49.71662 26.16947 33.48742
[155] 35.44070 29.29919 37.97488 15.39897 36.03442 54.76770 36.51149
[162] 44.87383 32.60741 41.23537 40.74941 28.11992 38.27005 36.23579
[169] 42.39755 25.57382 58.16302 33.23465 44.05196 58.51683 42.62157
[176] 59.05065 38.80158 28.75233 42.52488 32.05593 31.75240 41.07911
[183] 19.41156 34.44780 37.53963 25.71746 37.67745 33.60259 27.64901
[190] 42.55252 44.01244 39.47851 39.72053 27.21078 40.88741 36.13032
[197] 41.89054 45.96969 30.13897 35.42951mean_200 <- mean(samp)
mean_200[1] 35.87853sd_200 <- sd(samp)
sd_200[1] 8.17609hist(samp)
qqnorm(samp)
qqline(samp)
The distribution of the sample is normal and unimodal. The measure of center (mean) is 35.87853 and standard deviation is 8.18.
# Your code here
B <- 10^4
bsm <- numeric(B)
for(i in 1:B){
bsm[i] <- mean(sample(samp, 200, replace = TRUE))
}
hist(bsm)
mean_boot_200 <- mean(bsm)
sd_boot_200 <- sd(bsm)
qqnorm(bsm)
qqline(bsm)
The bootstrap distribution is normal and unimodal. The mean and standard error of the bootstrap is mean = 35.88 and sd = 0.578.
# Your code here
OD <- matrix(data = c(36, 36, mean_200, mean_boot_200, 8, 8/sqrt(200), sd_200, sd_boot_200), nrow = 4, byrow = FALSE)
dimnames(OD) = list(c("Population", "Sample Distribution", "Sample", "Bootstrap"),c("Mean", "SD"))
ODT <- as.table(OD)
knitr::kable(ODT, caption = "$n=200$")| Mean | SD | |
|---|---|---|
| Population | 36.00000 | 8.0000000 |
| Sample Distribution | 36.00000 | 0.5656854 |
| Sample | 35.87853 | 8.1760899 |
| Bootstrap | 35.87956 | 0.5815149 |
Parts a-d for \(n = 10\):
The distribution is exactly normal. The mean is 36 and the standard error is 2.5298221
# Your code here
set.seed(13)
sam <- rnorm(10, 36, 8)
sam [1] 40.43462 33.75782 50.20131 37.49856 45.14021 39.32421 45.83605
[8] 37.89344 33.07694 44.84115mean_10 <- mean(sam)
sd_10 <- sd(sam)
hist(sam)
qqnorm(sam)
qqline(sam)
# Your code here
set.seed(31)
B <- 10^4
bs <- numeric(B)
for(i in 1:B){
bs[i] <- mean(sample(sam, 10, replace = TRUE))
}
hist(bs)
mean_boot_10 <- mean(bs)
sd_boot_10 <- sd(bs)
qqnorm(bs)
qqline(bs)
# Your code here
OD <- matrix(data = c(36, 36, mean_10, mean_boot_10, 8, 8/sqrt(10), sd_10, sd_boot_10), nrow = 4, byrow = FALSE)
dimnames(OD) = list(c("Population", "Sample Distribution", "Sample", "Bootstrap"),c("Mean", "SD"))
ODT <- as.table(OD)
knitr::kable(ODT, caption = "$n=10$")| Mean | SD | |
|---|---|---|
| Population | 36.00000 | 8.000000 |
| Sample Distribution | 36.00000 | 2.529822 |
| Sample | 40.80043 | 5.575858 |
| Bootstrap | 40.78263 | 1.665502 |
Parts a-d for \(n = 50\):
The distribution is exactly normal. The mean is 36 and the standard error is 1.1313708
# Your code here
set.seed(13)
sa <- rnorm(50, 36, 8)
sa [1] 40.43462 33.75782 50.20131 37.49856 45.14021 39.32421 45.83605
[8] 37.89344 33.07694 44.84115 27.25125 39.69497 25.11212 21.15178
[15] 32.48116 34.44842 47.17145 36.80531 35.08449 41.61780 38.10034
[22] 50.68931 38.85922 27.63672 40.96147 37.19484 24.32547 19.78365
[29] 27.54434 30.17485 35.93431 42.78238 32.93207 31.78791 33.81419
[36] 31.15407 33.33706 34.06770 29.09780 29.22423 36.80272 48.72027
[43] 40.53196 48.91584 32.25080 30.19119 27.81329 20.49748 38.21718
[50] 47.26683mean_50 <- mean(sa)
sd_50 <- sd(sa)
hist(sa)
qqnorm(sa)
qqline(sa)
# Your code here
set.seed(31)
B <- 10^4
n <- numeric(B)
for(i in 1:B){
n[i] <- mean(sample(sa, 50, replace = TRUE))
}
hist(n)
mean_boot_50<-mean(n)
sd_boot_50 <-sd(n)
qqnorm(n)
qqline(n)
# Your code here
OD <- matrix(data = c(36, 36, mean_200, mean_boot_200, 8, 8/sqrt(50), sd_50, sd_boot_50), nrow = 4, byrow = FALSE)
dimnames(OD) = list(c("Population", "Sample Distribution", "Sample", "Bootstrap"),c("Mean", "SD"))
ODT <- as.table(OD)
knitr::kable(ODT, caption = "$n=50$")| Mean | SD | |
|---|---|---|
| Population | 36.00000 | 8.000000 |
| Sample Distribution | 36.00000 | 1.131371 |
| Sample | 35.87853 | 7.800473 |
| Bootstrap | 35.87956 | 1.092309 |
Your Answer here. As the sample size of the distribution decreases, the bootstrap distribution grows and then shrinks. At n = 200, the frequency is over 1200 and the qqplot Sample Quantiles from 34 to 38. At n = 50, the frequency is over 2000 and the qqplot Sample Quantiles from 36 to 46. At n = 10, the frequency is back at 1500, with a qqplot Sample Quantiles from 32 to 40. ————
- We investigate the bootstrap distribution of the median. Create random samples of size \(n\) for various \(n\) and bootstrap the median. Describe the bootstrap distribution. Change the sample sizes to 36 and 37; 200 and 201; 10,000 and 10,001. Note the similarities/dissimilarities, trends and so on. Why does the parity of the sample size matter?
set.seed(31)
ne <- 14 # n even
no <- 15 # n odd
wwe <- rnorm(ne) # draw random sample of size ne
wwo <- rnorm(no) # draw random sample of size no
N <- 10^4
even.boot <- numeric(N) # save space
odd.boot <- numeric(N)
for (i in 1:N)
{
x.even <- sample(wwe, ne, replace = TRUE)
x.odd <- sample(wwo, no, replace = TRUE)
even.boot[i] <- median(x.even)
odd.boot[i] <- median(x.odd)
}
Median <- c(even.boot, odd.boot)
Parity <- rep(c("n = 14", "n = 15"), each = N)
DF <- data.frame(Median = Median, Parity = Parity)
ggplot(data = DF, aes(x = Median)) +
geom_histogram(fill = "lightblue", color = "black") +
theme_bw() +
facet_grid(Parity ~.)
Figure 1: Histograms of bootstrapped median values
set.seed(31)
# Your code here
ne <- 36 # n even
no <- 37 # n odd
wwe <- rnorm(ne) # draw random sample of size ne
wwo <- rnorm(no) # draw random sample of size no
N <- 10^4
even.boot <- numeric(N) # save space
odd.boot <- numeric(N)
for (i in 1:N)
{
x.even <- sample(wwe, ne, replace = TRUE)
x.odd <- sample(wwo, no, replace = TRUE)
even.boot[i] <- median(x.even)
odd.boot[i] <- median(x.odd)
}
Median <- c(even.boot, odd.boot)
Parity <- rep(c("n = 36", "n = 37"), each = N)
DF <- data.frame(Median = Median, Parity = Parity)
ggplot(data = DF, aes(x = Median)) +
geom_histogram(fill = "lightblue", color = "black") +
theme_bw() +
facet_grid(Parity ~.)
Figure 2: Histograms of bootstrapped median values
set.seed(31)
# Your code here
ne <- 200 # n even
no <- 201 # n odd
wwe <- rnorm(ne) # draw random sample of size ne
wwo <- rnorm(no) # draw random sample of size no
N <- 10^4
even.boot <- numeric(N) # save space
odd.boot <- numeric(N)
for (i in 1:N)
{
x.even <- sample(wwe, ne, replace = TRUE)
x.odd <- sample(wwo, no, replace = TRUE)
even.boot[i] <- median(x.even)
odd.boot[i] <- median(x.odd)
}
Median <- c(even.boot, odd.boot)
Parity <- rep(c("n = 14", "n = 15"), each = N)
DF <- data.frame(Median = Median, Parity = Parity)
ggplot(data = DF, aes(x = Median)) +
geom_histogram(fill = "lightblue", color = "black") +
theme_bw() +
facet_grid(Parity ~.)
Figure 3: Histograms of bootstrapped median values
set.seed(31)
# Your code here
ne <- 1000 # n even
no <- 1001 # n odd
wwe <- rnorm(ne) # draw random sample of size ne
wwo <- rnorm(no) # draw random sample of size no
N <- 10^4
even.boot <- numeric(N) # save space
odd.boot <- numeric(N)
for (i in 1:N)
{
x.even <- sample(wwe, ne, replace = TRUE)
x.odd <- sample(wwo, no, replace = TRUE)
even.boot[i] <- median(x.even)
odd.boot[i] <- median(x.odd)
}
Median <- c(even.boot, odd.boot)
Parity <- rep(c("n = 1000", "n = 1001"), each = N)
DF <- data.frame(Median = Median, Parity = Parity)
ggplot(data = DF, aes(x = Median)) +
geom_histogram(fill = "lightblue", color = "black") +
theme_bw() +
facet_grid(Parity ~.)
Figure 4: Histograms of bootstrapped median values
Your answer: At n = 14 & 15, the median values stay towards 0 very much. At n = 36 & 37, The median sticks to 0, however, the frequency goes as far as -1 to 0.5 (this is a shorter distance of values than 14 and 15). At n = 200 & 201, The distribution gets even smaller, from -0.2 to 0.4. At n = 1000 & 1001, The distribution is only from -0.2 to 0.1. As the n values grow larger, the distribution grows smaller. Parity matters in sample sizes because it allows each graph to be of equal quality, meaning that the comparison of graphs by n values can be appropriate since the sample sizes are the same.
[Becca] 3. Import the data from data set Bangladesh. In addition to arsenic concentrations for 271 wells, the data set contains cobalt and chlorine concentrations.
(a) Conduct EDA on the chlorine concentrations and describe the salient features.
(b) Bootstrap the mean.
(c) Find and interpret the 95% bootstrap percentile confidence interval.
(d) What is the bootstrap estimate of the bias? What fraction of the bootstrap standard error does it represent?Bangladesh <- read.csv("http://www1.appstate.edu/~arnholta/Data/Bangladesh.csv")The Chlorine variable has some missing values. The following code will remove these entries:
chlorine <- subset(Bangladesh, select = Chlorine, subset = !is.na(Chlorine), drop = TRUE)Your answers:
chlplot <- ggplot(Bangladesh, aes(x=Chlorine))
chlplot + geom_histogram(aes(y = ..density..))
chlplot1 <- ggplot(Bangladesh, aes(sample=Chlorine))
chlplot1 + stat_qq()
mean(chlorine)[1] 78.08401sd(chlorine)[1] 210.0192The mean is 78.1 and the standard deviation is 210.0.
# Your code here
N = 10^4
bmean <- numeric(N)
for(i in 1:N){
bmean[i] <- mean(sample(chlorine, length(chlorine), replace=TRUE))
}
mean(bmean)[1] 78.33956sd(bmean)[1] 12.79261mean(bmean) - mean(chlorine)[1] 0.2555496The bootstrap mean is 78.0, the bootstrap standard deviation is 12.8. It has a bias of 0.04.
# Your code here
chlorine.quantile <- quantile(bmean, prob = c(.025, .975))
chlorine.quantile 2.5% 97.5%
55.23359 104.54672 We are 95% sure that the chlorine concetration falls between 55 and 105.
abs((mean(bmean) - mean(chlorine)) / sd(bmean))*100[1] 1.997634The bootstrap distribution has a bias of 0.04. This is 0.27% of the standard error.
- Consider Bangladesh chlorine (concentration). Bootstrap the trimmed mean (say, trim the upper and lower 25%) and compare your results with the usual mean (previous exercise).
Your answer:
# Your code here
sim = 10^4
trimmean <- numeric(sim)
for(i in 1:sim){
trimmean[i] <- mean(sample(chlorine, length(chlorine), replace=TRUE), trim = 0.25)
}
mean(trimmean)[1] 17.86695The bootstrapped trim mean is significantly lower than the usual mean.
The data set
FishMercurycontains mercury levels (parts per million) for 30 fish caught in lakes in Minnesota.Create a histogram or boxplot of the data. What do you observe?
Bootstrap the mean and record the bootstrap standard error and the 95% bootstrap percentile interval.
Remove the outlier and bootstrap the mean of the remaining data. Record the bootstrap standard error and the 95% bootstrap percentile interval.
What effect did removing the outlier have on the bootstrap distribution, in particular, the standard error?
FishMercury <- read.csv("http://www1.appstate.edu/~arnholta/Data/FishMercury.csv")
head(FishMercury) Mercury
1 1.870
2 0.160
3 0.088
4 0.160
5 0.145
6 0.099Your answers:
# Your code here
Mercury <- FishMercury$Mercury
hist(Mercury)
Note that there is one value (1.87) very far removed from the rest of the values.
- Bootstraping the mean and record the bootstrap standard error and 95% bootstrap percentile interval
n <- length(Mercury)
R <- 10^4
mercury.mean <- numeric(R)
for(i in 1:R){
x <- sample(Mercury, n, replace = TRUE)
mercury.mean[i] <- mean(x)
}
hist(mercury.mean, main = "Bootstrap distribution of mean")
abline(v = mean(mercury.mean), col = "red")
dev.new()
qqnorm(mercury.mean)
qqline(mercury.mean)
mean(mercury.mean)[1] 0.1816243BB <- mean(mercury.mean) - mean(Mercury) # BIAS
BB #Bootstrap Bias[1] -0.00024234SE <- sd(mercury.mean)
SE # Standard Error (Why am I getting a negative number?)[1] 0.05752721quantile(mercury.mean, c(0.025, 0.975)) # I am 95% sure that the value of the booststrapping Percentile Confidence interval is [0.1122, 0.3075]. 2.5% 97.5%
0.1120325 0.3063333 - Remove the outlier and bootstrap the mean of the remaining data. Record the bootstrap standard error and the 95% bootstrap percentile interval.
# What are the outliers How do I remove them?
mean(mercury.mean)[1] 0.1816243mean(mercury.mean) - mean(Mercury)[1] -0.00024234sd(mercury.mean)[1] 0.05752721quantile(mercury.mean, c(0.025, 0.975)) 2.5% 97.5%
0.1120325 0.3063333 In section 3.3, we performed a permutation test to determine if men and women consumed, on average, different amounts of hot wings.
Bootstrap the difference in means and describe the bootstrap distribution.
Find a 95% bootstrap percentile confidence interval for the difference of means and give a sentence interpreting this interval.
How do the bootstrap and permutation distribution differ?
BeerWings <- read.csv("http://www1.appstate.edu/~arnholta/Data/Beerwings.csv")
head(BeerWings) ID Hotwings Beer Gender
1 1 4 24 F
2 2 5 0 F
3 3 5 12 F
4 4 6 12 F
5 5 7 12 F
6 6 7 12 FYour answers:
female<- c(4,5,5,6,7,7,8,9,11,12,13,13,14,14)
male<-c(7,8,8,11,13,13,14,16,16,17,17,18,18,21,21)
W<- 10^4
dm <-numeric(W)
for (i in 1:W){
female_W <- sample(female,size= 14,replace=TRUE)
male_W<-sample(male,size=15, replace=TRUE)
dm[i] <- mean(male_W)-mean(female_W)
}
mean(dm)[1] 5.417626sd(dm)[1] 1.458709hist(dm)
The distribution of the differences in averages of the amount of hot wings between males and females is normal. The mean is 5.4 and standard deviation is 1.45.
female<- c(4,5,5,6,7,7,8,9,11,12,13,13,14,14)
male<-c(7,8,8,11,13,13,14,16,16,17,17,18,18,21,21)
W<- 10^4
mean_diff<-rep(NA,W)
for (i in 1:W){
female_W<-sample(female,replace=TRUE)
male_W<-sample(male, replace=TRUE)
mean_diff[i]<-mean(male_W)-mean(female_W)
}
quantile(mean_diff, prob= c(.025,.975)) 2.5% 97.5%
2.495119 8.209524 hist(mean_diff)
mean(mean_diff)[1] 5.371824sd(mean_diff)[1] 1.462362We are 95% confident that the difference in average consumed hot wings between men and women are 2.51 and 8.23.
- The bootstrap and permutation distributions do not differ all that much. The mean and sd of the normal permutatoin distribution are relatively the same as the bootstrap distribution. The mean is a little less and the standard deviation is a little more in the permutation distribution.
Import the data from
Girls2004(see Section 1.2).Perform some exploratory data analysis and obtain summary statistics on the weight of baby girls born in Wyoming and Arkansas (do seperate analyses for each state).
Bootstrap the difference in means, plot the distribution, and give the summary statistics. Obtain a 95% bootstrap percentile confidence interval and interpret this interval.
What is the bootstrap estimate of the bias? What fraction of the bootstrap standard error does it represent?
Conduct a permutation test to calculate the difference in mean weights and state your conclusion?
For what population(s), if any does this calculation hold? Explain?
Girls2004 <- read.csv("http://www1.appstate.edu/~arnholta/Data/Girls2004.csv")
head(Girls2004) ID State MothersAge Smoker Weight Gestation
1 1 WY 15-19 No 3085 40
2 2 WY 35-39 No 3515 39
3 3 WY 25-29 No 3775 40
4 4 WY 20-24 No 3265 39
5 5 WY 25-29 No 2970 40
6 6 WY 20-24 No 2850 38Your answers:
# Your code here
# Part a.
Girls2004 %>%
group_by(State) %>%
summarize(mean(Weight), sd(Weight), n = n())# A tibble: 2 x 4
State `mean(Weight)` `sd(Weight)` n
<fctr> <dbl> <dbl> <int>
1 AK 3516.35 578.8336 40
2 WY 3207.90 418.3184 40# Your code here
state_WY <- subset(Girls2004, select = Weight, subset = State == "WY", drop = TRUE)
state_AK <- subset(Girls2004, select = Weight, subset = State == "AK", drop = TRUE)
obsDiff <- mean(state_WY) - mean(state_AK)
B <- 10^4
states.diff.mean <- numeric(B)
set.seed(5)
for (i in 1:B){
WY_sample <- sample(state_WY, size = sum(!is.na(state_WY)), replace = TRUE)
AK_sample <- sample(state_AK, size = sum(!is.na(state_AK)), replace = TRUE)
states.diff.mean[i] <- mean(WY_sample) - mean(AK_sample)
}
hist(states.diff.mean)
mean(states.diff.mean)[1] -308.597sd(states.diff.mean)[1] 111.1064BCI <- quantile(states.diff.mean, prob = c(0.025, 0.975))
BCI 2.5% 97.5%
-525.82812 -88.09937 # Your code here
DF <- Girls2004 %>%
group_by(State) %>%
summarize(M = mean(Weight), sd(Weight), n = n()) %>%
summarize(thetahat = M[1] - M[2])
DF# A tibble: 1 x 1
thetahat
<dbl>
1 308.45bias <- mean(states.diff.mean) - mean(DF$thetahat)
bias[1] -617.047frac <- bias/sd(states.diff.mean)
frac[1] -5.553658# Your code here
norep <- Girls2004 %>%
group_by(State == "AK", State == "WY")
P <- 10^4
weights.diff.mean <- numeric(P)
set.seed(4)
for (i in 1:P){
index <- sample(length(norep$Weight), size = length(norep$Weight[norep$State == "AK"]), replace = FALSE)
weights.diff.mean[i] <- mean(norep$Weight[index] - norep$Weight[-index])
}
hist(weights.diff.mean, freq=FALSE,
main = "Permutation Distribution \n (Figure 7)",
xlab = substitute(paste(bar(x)[1],"*", - bar(x)[2],"*")),
col = "lightblue")
pvalue <- (sum(weights.diff.mean >= obsDiff) + 1)/(P + 1)
pvalue[1] 0.9957004The permutation test can be shown in the graph above (figure 7). The pvalue is .99, therefore we accept that there is difference in weights of newborns by state.
- The baby population for Arkansas has a higher weight in grams on average than Wyoming. This conclusion is towards mothers in the two states, Wyoming and Arkansas.
- Do chocolate and vanilla ice creams have the same number of calories? The data set
IceCreamcontains calorie information for a sample of brands of chocolate and vanilla ice cream. Use the bootstrap to determine whether or not there is a difference in the mean number of calories.
IceCream <- read.csv("http://www1.appstate.edu/~arnholta/Data/IceCream.csv")
head(IceCream) Brand VanillaCalories VanillaFat VanillaSugar ChocolateCalories
1 Baskin Robbins 260 16.0 26.0 260
2 Ben & Jerry's 240 16.0 19.0 260
3 Blue Bunny 140 7.0 12.0 130
4 Breyers 140 7.0 13.0 140
5 Brigham's 190 12.0 17.0 200
6 Bulla 234 13.5 21.8 266
ChocolateFat ChocolateSugar
1 14 31.0
2 16 22.0
3 7 14.0
4 8 16.0
5 12 18.0
6 15 22.6Your answer:
IceCream <- read.csv("http://www1.appstate.edu/~arnholta/Data/IceCream.csv")
head(IceCream) Brand VanillaCalories VanillaFat VanillaSugar ChocolateCalories
1 Baskin Robbins 260 16.0 26.0 260
2 Ben & Jerry's 240 16.0 19.0 260
3 Blue Bunny 140 7.0 12.0 130
4 Breyers 140 7.0 13.0 140
5 Brigham's 190 12.0 17.0 200
6 Bulla 234 13.5 21.8 266
ChocolateFat ChocolateSugar
1 14 31.0
2 16 22.0
3 7 14.0
4 8 16.0
5 12 18.0
6 15 22.6c <- IceCream$ChocolateCalories
v <- IceCream$VanillaCalories
d <- c-v
mean.diff <- mean(d)
N = 10^4
boot.mean.diff <- numeric(N)
for(i in 1:N){
boot.mean.diff[i] <- mean(sample(d,length(d),replace=TRUE))
}
CI <- quantile(boot.mean.diff, prob = c(.05, .95))
CI 5% 95%
4.076923 10.794872 boot.mean.diff [1] 8.1025641 5.7435897 8.4102564 5.1025641 10.5384615 6.5897436
[7] 11.0769231 6.8461538 2.9230769 5.6666667 5.2051282 8.0512821
[13] 6.4871795 9.7948718 6.7948718 6.9230769 11.1794872 6.8205128
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[1387] 7.7435897 9.5128205 6.1025641 7.2307692 7.6410256 7.1538462
[1393] 6.5897436 14.1282051 8.7692308 4.7948718 5.4358974 6.7435897
[1399] 8.0256410 8.3846154 6.4615385 9.2564103 5.2564103 7.0512821
[1405] 7.4615385 9.9487179 7.8461538 7.2564103 8.0769231 8.8717949
[1411] 5.5641026 7.0512821 6.1794872 6.6923077 3.9230769 8.2564103
[1417] 9.8717949 6.4358974 10.2051282 7.7435897 5.6410256 5.7435897
[1423] 9.0256410 9.4871795 6.0512821 11.2051282 6.9230769 6.5897436
[1429] 8.5128205 10.2051282 3.6153846 5.1794872 5.5897436 8.9487179
[1435] 4.9743590 6.4102564 5.8461538 3.4358974 9.0512821 7.4358974
[1441] 1.6410256 7.8205128 7.1282051 8.9230769 5.0769231 5.2820513
[1447] 8.6153846 5.4358974 5.9230769 6.2820513 7.6153846 8.4871795
[1453] 8.5128205 4.7179487 5.8717949 7.4102564 7.0256410 5.5384615
[1459] 9.3846154 4.2820513 8.4871795 6.4615385 2.7692308 12.1794872
[1465] 6.2307692 7.0256410 7.6410256 5.3589744 7.6153846 8.8974359
[1471] 8.9487179 9.9743590 9.0256410 7.7179487 4.3333333 5.9230769
[1477] 6.9487179 6.0769231 8.1025641 7.4358974 6.8461538 6.1794872
[1483] 8.4871795 2.6666667 4.7692308 8.3589744 3.5384615 9.1538462
[1489] 3.2307692 11.4102564 2.5641026 8.2307692 3.4102564 8.0512821
[1495] 6.6923077 5.3333333 5.4358974 5.6410256 8.4358974 7.3076923
[1501] 8.8717949 8.1282051 6.3076923 7.4358974 11.0256410 5.0000000
[1507] 9.8717949 7.3076923 7.5641026 3.9487179 8.9230769 6.9230769
[1513] 5.7692308 6.5897436 5.8205128 3.8717949 8.1025641 7.8974359
[1519] 8.6153846 6.7435897 11.4871795 10.8717949 5.3333333 7.5128205
[1525] 4.8461538 6.8974359 5.9743590 4.9230769 6.2307692 5.0769231
[1531] 8.0256410 8.8461538 6.3076923 3.7948718 6.7692308 7.2820513
[1537] 6.7435897 9.1282051 3.3076923 10.5128205 8.9743590 8.5128205
[1543] 9.9230769 8.2307692 5.0000000 2.3846154 2.6923077 11.8461538
[1549] 5.1538462 12.2564103 8.0256410 8.0000000 6.3589744 9.6666667
[1555] 6.7435897 7.4615385 4.7435897 4.0512821 3.5384615 8.1282051
[1561] 10.0000000 7.0000000 9.0000000 7.7435897 4.3333333 6.4102564
[1567] 10.2820513 8.5897436 7.2564103 6.0769231 5.6153846 5.7948718
[1573] 4.9230769 4.5384615 8.5128205 6.9487179 4.8205128 11.6666667
[1579] 9.0512821 9.2564103 6.1538462 9.3333333 8.2051282 9.1794872
[1585] 8.7692308 8.9230769 6.0256410 8.5128205 4.0256410 6.1794872
[1591] 7.3846154 10.8461538 5.2051282 7.6153846 7.3589744 11.1282051
[1597] 7.6666667 9.3589744 7.0512821 8.4358974 7.9487179 8.1282051
[1603] 6.4102564 9.4102564 9.1538462 8.9487179 5.0512821 8.1025641
[1609] 3.9743590 8.1794872 4.2307692 8.0512821 7.8974359 11.4615385
[1615] 6.8461538 7.4615385 7.9230769 6.2820513 6.4102564 8.0512821
[1621] 10.2051282 7.0256410 7.7435897 7.5384615 9.3846154 3.5897436
[1627] 5.6923077 8.8205128 5.2820513 6.3846154 4.7948718 9.6410256
[1633] 9.5384615 5.9487179 9.0769231 7.5641026 4.8461538 8.1794872
[1639] 6.6923077 5.3076923 6.6153846 6.7948718 6.8974359 7.5384615
[1645] 5.1794872 8.9487179 4.8205128 7.9743590 8.6923077 10.4102564
[1651] 9.0000000 8.2051282 6.4871795 5.9487179 9.3076923 7.3333333
[1657] 5.5128205 9.2307692 9.2564103 7.3076923 3.6410256 8.3589744
[1663] 12.2051282 7.7692308 5.1282051 8.3333333 7.7692308 6.2820513
[1669] 6.5897436 7.6410256 4.1282051 7.6923077 5.8974359 5.4615385
[1675] 5.0769231 8.7948718 7.3076923 6.4615385 8.0000000 9.4358974
[1681] 9.7692308 6.7692308 7.4102564 6.1538462 11.1538462 5.7692308
[1687] 6.4615385 6.1025641 10.4615385 9.3589744 11.8205128 10.6923077
[1693] 8.0000000 7.2051282 10.6923077 7.1025641 5.4358974 5.6153846
[1699] 9.2820513 8.1538462 9.6410256 3.2307692 4.1538462 7.4871795
[1705] 11.4358974 8.5384615 7.4358974 9.7948718 6.2051282 8.3589744
[1711] 7.0256410 8.5897436 6.1794872 8.6923077 8.5897436 11.1025641
[1717] 6.1794872 6.0256410 3.1025641 7.3333333 12.1794872 9.0256410
[1723] 8.7179487 8.5384615 9.8205128 9.5897436 5.3076923 6.5641026
[1729] 7.4102564 6.9743590 6.8974359 6.2051282 6.1025641 6.0512821
[1735] 6.5128205 8.0769231 9.2051282 8.7692308 9.2307692 6.9487179
[1741] 5.8974359 6.3846154 6.1538462 8.6666667 6.5897436 5.3846154
[1747] 5.8717949 4.6923077 9.8974359 7.6666667 5.5897436 6.7179487
[1753] 8.9230769 6.6153846 10.0000000 4.9487179 9.3846154 6.5128205
[1759] 4.0512821 5.0000000 9.6923077 7.7692308 11.8717949 7.6153846
[1765] 7.8205128 6.1794872 5.7435897 4.4358974 6.3076923 6.4358974
[1771] 7.3846154 8.3846154 11.8461538 4.7692308 6.8974359 5.4102564
[1777] 9.3333333 7.7179487 12.3846154 12.5384615 4.4358974 7.7435897
[1783] 7.6666667 8.1282051 7.3589744 8.5641026 4.1282051 5.8461538
[1789] 8.5384615 8.6153846 11.0256410 7.4358974 7.6410256 7.0256410
[1795] 7.8974359 10.1794872 9.2820513 6.3846154 6.7435897 7.8461538
[1801] 7.4615385 7.3846154 9.1794872 6.2051282 6.2820513 6.8205128
[1807] 6.6153846 7.1794872 9.3076923 5.0769231 6.5897436 5.7692308
[1813] 5.4615385 5.9487179 6.1282051 9.8205128 6.6410256 2.1794872
[1819] 8.4358974 4.0769231 3.6410256 5.5128205 5.6153846 8.2051282
[1825] 4.6666667 6.5128205 9.3589744 7.0256410 8.3846154 6.6410256
[1831] 7.5128205 7.8461538 8.4102564 6.2564103 7.8205128 6.2307692
[1837] 6.3333333 9.5128205 8.0000000 9.8205128 8.6666667 5.2307692
[1843] 6.2051282 4.9487179 4.2051282 5.7435897 10.6153846 7.5641026
[1849] 11.3846154 5.2820513 6.3846154 13.2051282 5.8461538 6.3333333
[1855] 6.5128205 5.8461538 5.7179487 8.8974359 3.1025641 7.3846154
[1861] 7.3589744 8.9743590 8.6666667 3.7692308 10.1025641 6.5897436
[1867] 7.9743590 6.3076923 8.6410256 4.4871795 6.5641026 9.4871795
[1873] 9.8974359 8.8974359 7.3333333 7.8461538 5.1282051 3.1282051
[1879] 3.9487179 9.0769231 8.9487179 9.9743590 6.4615385 8.2307692
[1885] 5.9487179 10.2307692 6.6410256 8.0769231 11.1282051 8.3333333
[1891] 8.2051282 5.1025641 7.3589744 8.7948718 3.0256410 9.6666667
[1897] 6.5384615 6.8974359 8.7692308 6.9487179 10.0256410 8.8717949
[1903] 9.8461538 8.8461538 7.9230769 11.5641026 11.4615385 8.1282051
[1909] 11.3846154 9.4102564 7.5897436 8.7179487 8.6410256 7.0512821
[1915] 5.5897436 5.8205128 8.0000000 8.4615385 5.1025641 9.3076923
[1921] 5.2820513 6.4102564 9.0000000 10.7948718 6.6923077 5.8205128
[1927] 7.8974359 15.0000000 6.4871795 6.4871795 4.3333333 4.6410256
[1933] 4.3333333 7.2051282 11.7435897 6.3333333 7.0512821 8.5128205
[1939] 6.0769231 5.8205128 9.8461538 11.3589744 6.9487179 5.5897436
[1945] 6.8717949 11.5128205 7.0769231 7.7435897 5.6923077 7.8461538
[1951] 6.9230769 9.2307692 9.2307692 7.6153846 7.9230769 7.7179487
[1957] 3.6923077 8.8974359 10.9743590 10.6923077 5.8974359 8.7692308
[1963] 6.4615385 8.1538462 9.2051282 6.3333333 7.8205128 5.9230769
[1969] 7.9230769 7.0512821 8.0769231 11.5128205 6.6410256 9.5641026
[1975] 7.0000000 6.7948718 11.8717949 4.9487179 8.9230769 8.2564103
[1981] 9.7435897 5.2564103 10.4102564 5.0769231 9.7435897 4.1025641
[1987] 5.6923077 6.2564103 7.5128205 4.3333333 6.1538462 7.7435897
[1993] 6.0512821 5.2051282 5.9230769 11.1538462 7.0512821 7.7179487
[1999] 5.7179487 9.7692308 8.9743590 6.3333333 8.1282051 5.9230769
[2005] 10.6153846 6.6666667 9.6153846 7.8717949 10.6923077 9.7179487
[2011] 7.8205128 3.1282051 5.5897436 8.0769231 8.9743590 9.0512821
[2017] 6.0000000 4.1282051 4.9487179 8.3846154 4.0512821 10.3076923
[2023] 7.7692308 10.4102564 4.5128205 9.4358974 5.1538462 8.5641026
[2029] 3.3333333 3.1025641 7.4102564 5.5128205 8.7692308 6.1025641
[2035] 6.7435897 6.7692308 8.1794872 6.7435897 10.5128205 7.4615385
[2041] 6.0000000 4.6410256 5.1025641 7.0000000 6.6410256 8.9230769
[2047] 5.9230769 8.3076923 6.1794872 6.6153846 7.5384615 5.2307692
[2053] 6.9487179 6.2307692 10.6410256 10.1794872 5.2820513 6.7179487
[2059] 5.7435897 8.4871795 9.8461538 10.8717949 5.1538462 4.7435897
[2065] 4.2051282 8.0000000 6.4102564 6.4358974 12.1538462 5.0512821
[2071] 9.1282051 9.7435897 7.4102564 6.3589744 8.8974359 5.8461538
[2077] 5.8974359 7.5641026 5.7179487 8.6410256 7.2564103 9.1794872
[2083] 7.4871795 8.8974359 7.0769231 12.3333333 12.3076923 5.4102564
[2089] 7.4102564 7.9743590 8.1538462 6.1538462 8.4358974 9.2307692
[2095] 4.4615385 6.4358974 10.6153846 9.4871795 9.1282051 6.1025641
[2101] 7.4871795 9.3333333 4.7692308 7.8461538 6.2051282 6.9230769
[2107] 8.2307692 7.2051282 8.5384615 4.3076923 5.5128205 7.7692308
[2113] 8.2051282 5.8974359 5.2307692 8.6153846 5.9743590 10.0000000
[2119] 6.7948718 7.5128205 7.1025641 10.3589744 9.7948718 8.4102564
[2125] 6.5641026 8.3846154 9.2564103 9.5897436 3.0000000 11.2307692
[2131] 7.3846154 8.1538462 9.4102564 7.2307692 8.8974359 9.6410256
[2137] 10.2820513 7.3846154 4.7435897 7.5897436 8.8461538 8.9743590
[2143] 10.6923077 8.7179487 9.9743590 4.6923077 9.7435897 7.7692308
[2149] 4.8717949 7.2307692 3.8205128 9.0512821 6.4102564 6.9230769
[2155] 6.7435897 8.7692308 8.5384615 10.8717949 7.4871795 9.9743590
[2161] 6.8717949 5.8974359 7.0769231 6.6410256 6.0769231 8.6923077
[2167] 2.6153846 5.6666667 10.7179487 3.5384615 5.3589744 8.7692308
[2173] 4.6923077 9.5128205 7.9230769 9.5641026 8.4358974 7.8461538
[2179] 7.8974359 6.4615385 8.7692308 3.6153846 6.0256410 9.8717949
[2185] 7.1025641 9.5641026 5.2820513 4.7435897 10.4871795 6.4871795
[2191] 9.3076923 11.0000000 7.2051282 5.6410256 6.6923077 9.1025641
[2197] 8.0000000 6.2051282 9.1538462 9.5641026 8.1025641 6.5897436
[2203] 5.1794872 5.4871795 5.9230769 8.1794872 10.8205128 5.3846154
[2209] 5.7692308 7.2307692 7.7692308 6.2820513 6.9230769 5.0000000
[2215] 6.9487179 9.4871795 8.6153846 6.9743590 6.4358974 5.4358974
[2221] 6.9743590 5.0512821 10.4358974 8.1794872 5.5897436 8.2307692
[2227] 6.9487179 7.6153846 6.1025641 10.3846154 10.1538462 8.0000000
[2233] 6.2564103 10.1794872 7.7948718 11.8717949 5.9743590 6.9230769
[2239] 10.2051282 8.1282051 11.0512821 6.6923077 9.7692308 6.5641026
[2245] 9.5641026 6.3846154 2.3333333 5.4358974 4.6410256 8.6666667
[2251] 9.3846154 9.6410256 9.7692308 11.1025641 8.7948718 5.7179487
[2257] 7.4615385 8.5897436 9.0512821 9.2820513 6.6410256 6.3589744
[2263] 4.7435897 7.8717949 6.5384615 5.2051282 9.1538462 6.0769231
[2269] 6.0769231 8.4615385 6.5128205 8.9487179 4.5128205 8.2820513
[2275] 5.8205128 7.3589744 6.2564103 7.6410256 7.1538462 7.7692308
[2281] 7.7435897 9.6410256 6.9487179 7.4615385 8.6153846 6.7692308
[2287] 4.7692308 9.8205128 12.9230769 6.0769231 11.5384615 6.2307692
[2293] 5.2820513 8.6153846 3.8974359 2.4102564 7.7435897 7.6666667
[2299] 5.5641026 10.2307692 8.8717949 8.2307692 6.6410256 10.8717949
[2305] 5.1025641 9.0256410 6.5897436 9.6923077 5.8717949 9.7179487
[2311] 10.4615385 6.4102564 6.3589744 5.3076923 4.3846154 10.6666667
[2317] 6.4615385 7.8461538 7.9487179 9.8974359 6.2307692 9.2051282
[2323] 11.0512821 5.3333333 7.8461538 6.9743590 6.6410256 7.3846154
[2329] 8.7435897 8.4102564 3.7179487 6.2307692 6.3333333 5.6923077
[2335] 7.7692308 3.3076923 1.4871795 3.9487179 6.6410256 5.6923077
[2341] 6.2564103 7.3333333 3.6153846 6.3076923 5.0000000 7.9487179
[2347] 6.6923077 8.9487179 6.5641026 5.0256410 5.0256410 8.7948718
[2353] 3.8717949 8.8205128 4.8205128 9.4102564 6.7179487 8.3846154
[2359] 5.2307692 7.0256410 14.1794872 7.4358974 3.3076923 5.8974359
[2365] 8.3846154 9.0769231 9.1794872 7.8974359 3.7948718 9.7435897
[2371] 5.3076923 8.7435897 5.7435897 10.0512821 6.8205128 5.6153846
[2377] 3.8974359 11.0000000 7.5897436 4.9743590 8.8717949 4.6666667
[2383] 9.5384615 3.7948718 7.0512821 9.8205128 7.3076923 8.0512821
[2389] 8.5897436 8.9743590 7.7435897 8.1025641 5.9743590 9.4102564
[2395] 6.5641026 4.4102564 8.6666667 7.8974359 6.4102564 7.8717949
[2401] 6.4615385 4.8205128 7.6410256 4.2820513 6.0000000 8.2564103
[2407] 8.5897436 6.2307692 2.8205128 9.6153846 5.3589744 5.1025641
[2413] 7.5384615 3.7435897 8.3846154 7.2820513 6.5384615 6.1025641
[2419] 5.0000000 8.0000000 6.6153846 4.8717949 7.6666667 3.6666667
[2425] 4.4615385 5.6923077 6.3846154 10.2307692 7.2307692 10.1794872
[2431] 9.8974359 11.7692308 6.5128205 8.5384615 10.8717949 15.7179487
[2437] 8.7948718 6.9743590 9.1025641 8.4871795 8.9487179 7.6923077
[2443] 3.9487179 9.1538462 7.1794872 6.8461538 5.7948718 6.4358974
[2449] 9.6923077 8.0512821 10.6666667 6.4871795 7.7435897 6.4871795
[2455] 6.3589744 6.3589744 7.4102564 6.0769231 5.6923077 8.2564103
[2461] 8.5897436 4.0769231 6.1282051 10.3846154 6.1025641 6.7179487
[2467] 6.0256410 6.9487179 3.6410256 5.1538462 9.5384615 7.2820513
[2473] 4.9230769 5.9230769 5.9487179 8.4615385 4.0000000 9.8205128
[2479] 6.5897436 8.9743590 10.3333333 7.3333333 7.8461538 3.1538462
[2485] 6.8974359 4.6410256 7.2564103 7.1025641 10.2820513 6.8205128
[2491] 7.5128205 8.7692308 6.5897436 6.8717949 7.3846154 8.1282051
[2497] 7.2051282 6.3076923 8.2564103 7.3333333 3.3333333 3.3076923
[2503] 6.6923077 7.9743590 5.7435897 4.4615385 8.3333333 6.8205128
[2509] 3.9743590 9.4871795 7.6923077 3.1794872 10.5641026 9.2564103
[2515] 7.2307692 7.0512821 7.4871795 7.9230769 7.8974359 8.3846154
[2521] 7.9487179 10.3589744 6.9743590 4.4615385 7.6153846 6.1538462
[2527] 11.6153846 6.4615385 6.1282051 4.8974359 6.1794872 7.4358974
[2533] 8.5897436 7.8974359 9.8974359 8.3333333 4.5128205 8.3846154
[2539] 9.8717949 4.7435897 8.5384615 5.8205128 5.1538462 11.6153846
[2545] 8.9230769 3.3589744 6.4871795 8.7692308 8.6666667 8.2564103
[2551] 6.2051282 6.8461538 8.4615385 9.8717949 3.7435897 8.6410256
[2557] 8.1025641 8.2307692 2.8974359 7.5128205 5.5128205 5.9230769
[2563] 6.4102564 9.2564103 9.6923077 7.8461538 5.0256410 6.8974359
[2569] 7.0769231 8.0256410 8.3333333 2.7692308 7.0512821 6.0256410
[2575] 4.7692308 7.8717949 5.0256410 8.1282051 5.7435897 5.7435897
[2581] 4.4615385 5.8461538 1.7179487 7.4615385 8.4358974 8.2564103
[2587] 10.2307692 11.2820513 4.1282051 6.4615385 7.0000000 6.1538462
[2593] 5.2307692 5.8974359 7.0000000 9.4871795 9.5384615 6.6666667
[2599] 4.4102564 10.2564103 4.7692308 7.8974359 9.9487179 6.2564103
[2605] 10.7435897 6.4102564 3.2051282 7.1282051 7.3589744 9.1282051
[2611] 7.2307692 7.9743590 8.7435897 6.2051282 9.8461538 3.3846154
[2617] 12.2564103 5.0000000 8.6153846 4.8461538 4.9487179 7.8974359
[2623] 7.4615385 9.3846154 10.7948718 6.1538462 7.3589744 7.3333333
[2629] 11.0000000 6.6153846 6.2564103 4.4615385 2.4358974 5.8717949
[2635] 6.0000000 6.5128205 8.2820513 6.0512821 6.5384615 6.4358974
[2641] 7.7179487 6.2307692 10.3846154 11.7179487 6.4615385 5.7692308
[2647] 7.6410256 9.3846154 3.4871795 8.4102564 7.0256410 5.7948718
[2653] 7.2051282 4.3333333 5.4358974 5.4615385 9.1794872 6.5897436
[2659] 7.1794872 8.6923077 8.4358974 9.3589744 6.3589744 7.3333333
[2665] 8.5128205 9.1538462 11.0769231 8.3589744 4.9230769 5.2307692
[2671] 7.5128205 7.9743590 8.7179487 4.2564103 8.6153846 13.6153846
[2677] 5.4102564 7.0256410 8.6666667 8.7179487 6.0000000 7.8974359
[2683] 10.3589744 7.9230769 4.4102564 9.0256410 3.4102564 7.3589744
[2689] 7.7948718 3.6410256 11.8974359 4.3333333 8.7948718 9.8205128
[2695] 6.9230769 6.1538462 8.8205128 6.8974359 6.5384615 10.3076923
[2701] 5.4615385 5.4102564 5.5128205 4.1538462 6.9487179 6.7948718
[2707] 7.7948718 8.9743590 4.6410256 1.7435897 5.3589744 7.2564103
[2713] 4.1794872 6.0769231 8.3076923 2.0000000 8.7435897 7.1025641
[2719] 6.8974359 6.2564103 8.0000000 7.8717949 7.9487179 6.1538462
[2725] 7.0769231 9.5384615 5.6923077 10.0769231 6.9230769 7.6666667
[2731] 7.7435897 7.1025641 7.7948718 6.7179487 8.5641026 6.5897436
[2737] 7.1025641 10.1538462 9.2051282 6.0769231 6.1538462 7.6153846
[2743] 4.0256410 5.0769231 6.2307692 8.0000000 10.7948718 7.7948718
[2749] 7.0769231 4.1538462 11.7948718 4.9743590 7.0769231 8.8461538
[2755] 7.5897436 9.7435897 9.0512821 7.7692308 7.0512821 10.5128205
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[2821] 8.3076923 3.5128205 8.8974359 6.1282051 4.1794872 10.0000000
[2827] 9.4871795 8.7179487 8.1282051 8.7948718 7.8461538 9.2564103
[2833] 8.0512821 7.9230769 6.6153846 5.0000000 8.4102564 5.9743590
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[3229] 11.7179487 6.0769231 7.7435897 11.8717949 11.1794872 8.1794872
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[3655] 9.3076923 10.4871795 6.7435897 4.8717949 8.8461538 11.0000000
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[3673] 10.1794872 7.7692308 4.8461538 6.8717949 10.0000000 6.3076923
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[3697] 8.7435897 4.8205128 6.4615385 4.4871795 5.1794872 7.0000000
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[3721] 3.5384615 3.6410256 8.6923077 6.6666667 7.4871795 3.0000000
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[4033] 6.0000000 7.7179487 7.0769231 5.4358974 10.1538462 7.0000000
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[4045] 1.7948718 8.3333333 8.2307692 8.8974359 3.8974359 6.3333333
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[4123] 9.4615385 6.8974359 7.9487179 9.1025641 4.4358974 5.8974359
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[4135] 9.2820513 6.1794872 8.3076923 6.3333333 2.2820513 7.7435897
[4141] 6.6666667 7.6666667 12.4615385 8.6153846 4.9487179 10.3076923
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[4153] 9.2820513 6.2820513 9.5384615 8.6410256 8.9230769 6.0512821
[4159] 7.7179487 7.3076923 8.4615385 9.4615385 9.3589744 5.4358974
[4165] 8.1025641 11.2051282 5.1025641 9.2051282 4.3589744 7.4358974
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[4183] 10.7948718 6.3589744 8.7692308 7.9743590 8.7179487 7.0512821
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[4207] 8.2051282 8.9230769 8.4615385 6.5897436 9.7435897 6.0256410
[4213] 5.5641026 6.6923077 7.4871795 6.6666667 4.7179487 9.0512821
[4219] 8.8974359 9.1794872 5.8205128 6.7179487 8.0256410 8.7179487
[4225] 8.9487179 15.6410256 9.4102564 6.5897436 11.4102564 5.5384615
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[4243] 4.5641026 8.6410256 4.5897436 7.0769231 7.6666667 9.5897436
[4249] 5.0769231 5.8974359 5.1794872 4.8461538 6.2307692 11.3076923
[4255] 7.7948718 6.0000000 10.1025641 6.3589744 10.3333333 4.4871795
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[4279] 4.2051282 6.1025641 3.7435897 6.3076923 8.2564103 5.1025641
[4285] 9.4871795 4.8717949 5.9230769 7.7948718 7.6923077 10.4871795
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[4315] 12.7435897 7.8717949 5.9230769 5.2820513 8.7435897 7.2051282
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[4399] 5.4358974 6.5897436 7.6410256 5.1282051 12.4358974 5.9743590
[4405] 5.3333333 6.8461538 7.6666667 7.4871795 6.5897436 9.0000000
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[4417] 9.4102564 5.4871795 8.5641026 6.7692308 10.5897436 6.1025641
[4423] 5.9230769 7.3846154 5.6666667 5.7435897 7.8461538 4.8717949
[4429] 5.2564103 6.6153846 6.9230769 7.8205128 5.5641026 5.8974359
[4435] 3.8205128 7.8205128 10.1794872 5.2051282 9.0000000 7.9230769
[4441] 6.2564103 5.4358974 9.3846154 7.4358974 7.1025641 7.0512821
[4447] 5.8974359 7.3846154 7.4358974 5.4615385 9.8205128 8.9487179
[4453] 6.6410256 5.8717949 9.3846154 9.0769231 11.1282051 3.2051282
[4459] 8.6666667 8.7435897 10.1025641 8.9230769 4.3076923 5.5128205
[4465] 7.2307692 5.0000000 5.4871795 7.9230769 10.8974359 8.8205128
[4471] 6.2307692 6.1282051 5.1794872 7.4358974 5.3076923 7.6153846
[4477] 8.2051282 3.8974359 5.0000000 6.2051282 6.4358974 10.5128205
[4483] 6.5897436 10.7948718 8.3333333 9.7179487 4.9230769 2.8205128
[4489] 8.8461538 8.1794872 4.8205128 9.3333333 8.9743590 9.2564103
[4495] 7.1794872 7.7179487 9.4871795 6.3589744 8.1538462 6.8717949
[4501] 13.7179487 5.1794872 6.6410256 8.6666667 7.1282051 5.4358974
[4507] 10.3846154 10.6666667 6.9743590 4.3076923 8.9487179 7.3333333
[4513] 4.7179487 7.1282051 9.3333333 4.6666667 5.0512821 8.4358974
[4519] 8.7435897 10.2051282 5.2307692 8.0769231 6.7692308 7.4871795
[4525] 2.8205128 6.2564103 7.3846154 5.0000000 10.3846154 3.9230769
[4531] 7.8717949 5.8461538 4.5384615 8.2051282 8.1025641 6.1282051
[4537] 7.6410256 6.7435897 9.3076923 7.3076923 5.1025641 4.9230769
[4543] 8.3076923 6.4358974 5.3589744 7.5641026 3.4615385 8.3076923
[4549] 8.9230769 5.9487179 5.7435897 8.5641026 5.6923077 6.7692308
[4555] 10.0512821 5.8717949 8.8717949 7.0256410 9.6410256 7.3589744
[4561] 5.3333333 8.9743590 8.2564103 2.1025641 8.8461538 4.8717949
[4567] 8.0256410 9.4102564 7.4615385 4.5384615 4.4871795 7.6153846
[4573] 7.5128205 6.2820513 6.6923077 8.1794872 7.4871795 9.6153846
[4579] 6.9743590 7.9230769 10.8974359 10.8461538 6.0769231 2.3846154
[4585] 9.2820513 6.1282051 9.3589744 6.8717949 9.9487179 5.6410256
[4591] 6.6923077 4.5897436 9.2307692 6.6410256 8.5128205 7.0000000
[4597] 11.0769231 8.8461538 9.5128205 6.6666667 6.7179487 11.1025641
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[4621] 4.6410256 7.2051282 4.7948718 9.1794872 9.1282051 11.8974359
[4627] 5.0769231 4.7435897 8.0256410 7.7692308 7.2820513 6.2564103
[4633] 6.5384615 7.3589744 8.7948718 4.1538462 8.3333333 7.5897436
[4639] 8.6153846 8.6153846 6.5641026 9.7435897 7.5384615 9.9230769
[4645] 9.7435897 7.7435897 4.6153846 10.3589744 7.9487179 7.7435897
[4651] 4.9230769 9.5128205 7.1538462 5.4615385 7.5897436 3.8717949
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[4675] 10.1538462 4.3846154 8.3076923 6.6923077 5.3333333 6.0512821
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[4705] 8.5641026 8.2051282 11.4615385 5.2051282 4.3846154 6.7435897
[4711] 9.7692308 4.3846154 6.7179487 8.1025641 4.1282051 9.7435897
[4717] 8.7948718 5.9230769 7.3333333 8.1794872 8.5384615 8.6666667
[4723] 7.1794872 7.7435897 7.4615385 9.4615385 4.5384615 6.0512821
[4729] 9.5641026 8.4102564 5.3589744 3.2051282 8.6153846 5.4358974
[4735] 8.6410256 8.1794872 7.7435897 10.0512821 5.5897436 8.3846154
[4741] 9.6666667 10.1794872 4.0000000 6.6153846 6.4615385 9.0000000
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[4759] 7.3333333 8.1025641 7.7435897 7.4358974 9.8205128 9.5128205
[4765] 9.0000000 9.1538462 10.7435897 8.2051282 8.8974359 9.7948718
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[4783] 6.3846154 6.2307692 10.5128205 11.4358974 9.0769231 5.1538462
[4789] 9.2820513 6.8717949 7.6153846 9.4102564 9.4871795 7.4615385
[4795] 5.8205128 6.2051282 8.6410256 5.0256410 8.3076923 6.0769231
[4801] 9.8461538 6.5641026 5.8461538 6.7692308 3.6923077 4.7435897
[4807] 12.7948718 9.0769231 6.5641026 9.1538462 8.0256410 4.7179487
[4813] 5.1538462 4.9743590 4.2564103 10.8974359 7.1025641 5.8974359
[4819] 9.1282051 3.6923077 5.1794872 3.9743590 7.0512821 13.1282051
[4825] 4.7948718 7.2307692 7.2820513 3.3846154 6.4871795 9.2564103
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[4837] 7.4615385 7.2051282 6.4102564 6.9230769 6.6153846 11.2820513
[4843] 8.4102564 8.2307692 11.1025641 6.8461538 8.3589744 5.3076923
[4849] 6.2820513 9.5897436 9.5897436 8.5128205 5.9230769 5.7435897
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[4861] 9.2051282 7.0769231 8.3076923 6.0000000 8.5384615 4.4358974
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[4945] 12.6410256 9.3076923 11.4871795 9.6410256 7.5641026 6.2820513
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[5011] 11.5128205 7.2051282 9.4615385 9.2564103 5.7179487 4.7948718
[5017] 7.9230769 5.6923077 9.3076923 6.3333333 7.0000000 10.2307692
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[5029] 5.8974359 7.2564103 6.0769231 7.0000000 7.5128205 7.2307692
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[5047] 9.8974359 7.4871795 5.5641026 5.9230769 5.1538462 8.2564103
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[5059] 9.1794872 11.8717949 6.1282051 9.1282051 3.7948718 7.5128205
[5065] 4.7179487 7.5384615 8.1282051 5.1538462 7.3589744 9.8461538
[5071] 7.6153846 10.3333333 8.1538462 6.6923077 6.7179487 9.7179487
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[5083] 11.6923077 6.5384615 5.1538462 8.4871795 7.3076923 4.8717949
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[5095] 10.0256410 4.9487179 8.2307692 4.5384615 7.5641026 9.8461538
[5101] 9.3333333 7.8461538 6.7179487 5.4871795 8.7692308 11.4871795
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[5119] 10.3589744 6.0769231 9.3076923 5.0512821 5.5641026 5.9487179
[5125] 4.7179487 8.9487179 12.0000000 6.2307692 6.9487179 7.5641026
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[5143] 8.1794872 8.0000000 6.5897436 5.0769231 11.5641026 9.1538462
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[5209] 6.3589744 9.3076923 8.9487179 7.1025641 8.9230769 11.1025641
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[5233] 7.5641026 5.0512821 6.6666667 10.0000000 8.2307692 6.8974359
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[5245] 9.8205128 6.3589744 7.8717949 5.1794872 10.8717949 5.2820513
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[5257] 11.8974359 8.9230769 11.3076923 8.0256410 7.2564103 5.3589744
[5263] 4.7692308 9.0000000 5.9743590 5.8974359 4.6410256 11.4102564
[5269] 4.8205128 9.0000000 8.4871795 3.7435897 6.9230769 11.3076923
[5275] 6.3589744 5.6923077 6.8461538 6.5128205 8.0000000 6.5384615
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[5293] 7.2051282 5.2307692 8.4871795 12.7435897 6.4871795 5.6666667
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[5311] 8.1025641 8.6410256 6.4615385 7.4615385 6.1538462 6.2051282
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[5329] 9.4871795 10.0769231 7.8717949 7.8717949 7.1282051 8.7435897
[5335] 7.1538462 7.4102564 5.0769231 10.1538462 5.5897436 6.0000000
[5341] 10.0512821 8.3589744 5.6666667 3.1538462 10.6410256 11.4358974
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[5353] 4.0000000 10.3076923 7.9743590 11.9230769 7.4871795 8.8461538
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[5365] 9.4358974 5.8205128 6.6153846 6.3589744 7.6666667 8.7948718
[5371] 7.1794872 7.4102564 8.1538462 8.9487179 5.6923077 6.9230769
[5377] 10.4871795 6.4102564 4.0000000 10.6666667 7.6923077 7.1025641
[5383] 6.8974359 8.4615385 12.9743590 5.1794872 7.7692308 6.9487179
[5389] 7.4871795 8.9743590 11.3846154 8.7948718 7.1794872 5.9230769
[5395] 5.6410256 5.4871795 7.4615385 6.7179487 7.9487179 6.1282051
[5401] 9.7435897 7.9230769 5.0512821 4.2307692 5.2820513 5.5641026
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[5413] 5.8717949 7.5384615 7.7435897 3.9743590 8.5128205 8.7948718
[5419] 6.0512821 8.1282051 7.1794872 8.3589744 7.4871795 9.2820513
[5425] 10.9487179 8.2564103 6.8461538 5.7948718 9.6153846 3.8205128
[5431] 9.0000000 8.8974359 5.2820513 6.6410256 7.1538462 6.2051282
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[5455] 6.7179487 6.3846154 7.9743590 9.3333333 9.6666667 9.3076923
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[6589] 7.4615385 9.4615385 4.3076923 6.3333333 11.5384615 10.0000000
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[8275] 3.6666667 10.0000000 7.8461538 4.6410256 7.4102564 11.0000000
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[8287] 7.6410256 7.7692308 8.0000000 5.2051282 2.6923077 7.2051282
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[8551] 7.0000000 9.3589744 8.6153846 8.3589744 9.3589744 12.8974359
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[8569] 6.7435897 4.8205128 8.0512821 4.6923077 8.7948718 4.3589744
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[8599] 12.1282051 8.0000000 12.7179487 6.8461538 8.7435897 6.0256410
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[8611] 12.9487179 7.4615385 10.4871795 6.3333333 5.0512821 7.1794872
[8617] 7.2307692 6.2051282 8.0769231 5.7948718 3.0000000 7.8205128
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[8629] 3.0769231 5.4615385 8.8717949 6.7435897 5.4615385 7.6153846
[8635] 9.2564103 9.4358974 4.6410256 7.7179487 6.5897436 7.4358974
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[8881] 6.1538462 5.1794872 5.3333333 8.0512821 9.5384615 7.3846154
[8887] 6.9487179 7.1025641 9.6410256 2.4102564 7.9230769 9.7179487
[8893] 7.2820513 12.0512821 1.3333333 7.9230769 8.3333333 6.9230769
[8899] 9.9487179 5.9487179 8.3076923 8.8461538 10.5128205 9.3333333
[8905] 7.6666667 7.3333333 5.9487179 4.1025641 6.6923077 6.2307692
[8911] 7.1538462 9.4871795 6.1794872 3.2820513 6.2820513 9.0512821
[8917] 12.0769231 7.6410256 6.3333333 4.5384615 9.4871795 8.8974359
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[8965] 4.7179487 5.7179487 9.2307692 8.4871795 11.0512821 11.3333333
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[8989] 1.0256410 8.9487179 5.5128205 9.1282051 7.5897436 7.3076923
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[9001] 12.2820513 8.3846154 8.1794872 5.6153846 6.5384615 8.3333333
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[9091] 5.7948718 4.6666667 9.1025641 9.3076923 7.3589744 6.7179487
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[9157] 7.2564103 7.7948718 6.4871795 11.3333333 5.5384615 10.3076923
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[9169] 11.3846154 9.1282051 7.0000000 8.2307692 12.1538462 6.6666667
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[9181] 8.2307692 7.3846154 5.4615385 3.4615385 4.8974359 5.7435897
[9187] 7.4615385 5.6666667 11.5384615 5.8974359 5.7179487 4.5128205
[9193] 9.0000000 8.3076923 4.7435897 6.2051282 8.4615385 8.3846154
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[9205] 9.2564103 6.7948718 8.9487179 6.3076923 6.6153846 8.8974359
[9211] 8.3846154 7.1794872 10.8974359 9.2564103 8.4102564 13.0000000
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[9247] 3.6923077 7.7692308 7.4871795 11.0000000 5.8974359 7.1282051
[9253] 10.8974359 5.4615385 4.6666667 9.6666667 8.6666667 9.9487179
[9259] 9.5128205 7.5128205 7.3076923 2.8717949 6.0000000 6.1538462
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[9511] 4.0256410 5.4871795 7.6923077 9.0256410 6.1538462 8.2564103
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[9535] 7.2307692 7.0256410 6.1538462 5.8974359 6.4358974 11.2307692
[9541] 5.4615385 7.2820513 7.1282051 8.6923077 6.0769231 9.1025641
[9547] 7.7435897 3.9743590 4.9230769 8.4615385 8.7948718 5.3333333
[9553] 2.0512821 6.7948718 7.8974359 3.4102564 4.6666667 3.4615385
[9559] 7.5897436 7.4871795 8.1282051 6.0769231 6.1538462 6.4358974
[9565] 7.4102564 4.9743590 9.8974359 4.2564103 5.0512821 9.8205128
[9571] 5.7948718 7.8974359 7.3589744 8.4871795 8.5128205 7.8461538
[9577] 10.4615385 4.1794872 6.0512821 9.6410256 9.5128205 9.2564103
[9583] 8.2307692 6.1282051 4.3846154 6.4358974 7.5384615 9.8205128
[9589] 8.8205128 6.5641026 6.6153846 8.6153846 8.6666667 5.8974359
[9595] 7.2307692 8.6666667 3.1282051 7.0000000 5.6153846 7.2051282
[9601] 7.7948718 7.6666667 7.7948718 6.2307692 7.0256410 8.2820513
[9607] 5.4615385 8.3589744 9.6666667 10.1282051 6.0256410 5.1794872
[9613] 8.5128205 1.9230769 6.4358974 7.0000000 6.5128205 6.8461538
[9619] 7.1282051 7.7948718 9.4615385 5.2820513 5.9487179 7.9230769
[9625] 5.9487179 9.9230769 6.0769231 5.7948718 8.0256410 7.3846154
[9631] 8.3076923 8.5128205 5.2820513 10.1025641 11.3589744 8.3076923
[9637] 11.7948718 8.2051282 8.0256410 5.1282051 8.4615385 4.3846154
[9643] 8.4615385 4.6923077 4.4358974 5.2820513 8.1025641 5.1025641
[9649] 6.5897436 11.1025641 9.5641026 7.4358974 5.5128205 8.7179487
[9655] 6.6923077 8.4102564 7.9743590 9.0512821 4.1282051 5.4102564
[9661] 7.8205128 7.6410256 5.5128205 11.9743590 10.1538462 5.5128205
[9667] 5.3589744 4.7692308 5.9743590 9.2564103 4.6410256 8.4102564
[9673] 4.2564103 4.5641026 9.4615385 6.1538462 7.7948718 9.4102564
[9679] 3.6666667 10.9487179 12.1025641 6.9230769 5.2564103 8.0512821
[9685] 7.7435897 11.6666667 4.9230769 0.7948718 6.5384615 6.5897436
[9691] 6.9743590 9.9487179 10.4871795 10.6153846 6.7435897 6.0512821
[9697] 5.6410256 4.8461538 6.8205128 3.6923077 7.8974359 9.2307692
[9703] 7.3333333 7.8205128 9.3846154 5.9487179 6.1794872 6.5641026
[9709] 7.7948718 9.4871795 6.1025641 4.8205128 6.2051282 5.7179487
[9715] 9.6923077 4.8717949 7.5641026 8.2820513 10.8205128 6.6923077
[9721] 9.3846154 5.6923077 5.3846154 6.1282051 5.9230769 11.1025641
[9727] 5.8974359 9.4358974 6.5897436 7.2820513 9.6153846 5.8205128
[9733] 3.0512821 8.6923077 8.0769231 9.9487179 6.1794872 7.2564103
[9739] 6.7435897 10.8461538 4.4358974 4.6153846 8.8974359 4.7948718
[9745] 9.4615385 8.0769231 6.4358974 5.3076923 9.3846154 8.0512821
[9751] 5.7435897 8.8974359 6.2820513 8.3589744 12.0256410 9.3846154
[9757] 4.6923077 3.7179487 7.6923077 5.7435897 11.4358974 4.7435897
[9763] 3.8205128 6.4358974 8.3846154 6.7948718 6.2307692 6.4358974
[9769] 7.3589744 7.7435897 6.3076923 8.1538462 8.4871795 8.3076923
[9775] 6.6410256 6.2820513 8.2051282 5.3846154 11.1282051 8.1794872
[9781] 8.5897436 7.1538462 9.4615385 7.5384615 6.0512821 8.0256410
[9787] 7.4102564 8.9487179 6.3333333 6.9743590 6.7435897 8.5897436
[9793] 9.1282051 8.8461538 8.8205128 7.2051282 3.9487179 10.5897436
[9799] 7.1282051 7.3333333 9.8974359 6.8205128 7.7692308 8.0256410
[9805] 6.3589744 8.4615385 6.6923077 5.3589744 8.3846154 4.9230769
[9811] 6.7948718 7.7692308 3.0256410 8.8974359 7.6410256 8.1282051
[9817] 8.6923077 5.5897436 6.5384615 6.5897436 4.6410256 4.4358974
[9823] 5.4615385 9.4358974 8.3333333 7.0256410 5.6666667 6.3846154
[9829] 10.9487179 4.1538462 10.4615385 4.5384615 7.5897436 8.0000000
[9835] 8.0769231 9.3333333 6.8717949 8.8717949 8.2564103 9.9487179
[9841] 4.1282051 7.4871795 8.2564103 9.8461538 8.0512821 7.7179487
[9847] 6.5384615 8.2051282 1.8717949 9.1538462 5.9743590 5.2307692
[9853] 6.6666667 6.4615385 7.1025641 4.5128205 5.9230769 9.6153846
[9859] 6.3333333 4.5897436 7.2820513 7.3589744 7.0256410 5.8461538
[9865] 6.8717949 7.4358974 8.5897436 5.3589744 8.3333333 8.5897436
[9871] 5.6923077 9.4615385 6.1794872 8.7435897 6.2820513 7.2307692
[9877] 4.8205128 7.3333333 10.3846154 8.6666667 5.7179487 9.7435897
[9883] 9.6410256 6.3846154 6.3846154 6.2307692 5.3076923 6.1538462
[9889] 9.4358974 12.2564103 10.2307692 6.7692308 6.7948718 9.0769231
[9895] 6.7692308 4.7692308 7.2307692 7.6410256 4.8974359 5.6153846
[9901] 7.9743590 9.4871795 8.5897436 6.8461538 7.7179487 7.5384615
[9907] 11.7179487 8.0000000 7.7435897 7.8717949 5.7435897 7.8205128
[9913] 2.9230769 6.8461538 5.4871795 5.2307692 7.6666667 3.6410256
[9919] 7.5384615 8.0769231 4.6153846 4.4871795 6.9743590 4.1025641
[9925] 5.4871795 6.9743590 5.9230769 10.3846154 4.1282051 7.4102564
[9931] 2.0769231 7.0256410 8.8974359 9.9743590 9.6666667 11.5384615
[9937] 5.3589744 5.7435897 4.0512821 8.1794872 4.5641026 7.1794872
[9943] 9.6923077 6.6153846 8.1282051 5.2051282 7.5897436 6.0256410
[9949] 8.2820513 7.7179487 4.8974359 7.6410256 5.3076923 11.8717949
[9955] 10.0256410 8.5384615 9.5897436 7.7692308 5.8205128 12.4102564
[9961] 6.2820513 7.5897436 9.2820513 5.1282051 5.9230769 6.3076923
[9967] 8.2051282 6.0000000 7.6410256 9.8717949 5.3076923 8.6153846
[9973] 9.4615385 8.0512821 9.7692308 7.0000000 7.0769231 5.2307692
[9979] 7.0769231 7.7179487 7.7179487 11.3589744 4.3333333 7.0000000
[9985] 8.2051282 6.1794872 3.4871795 4.0769231 7.9743590 6.6410256
[9991] 8.5897436 8.8205128 6.7948718 6.9487179 7.9743590 6.4615385
[9997] 5.6923077 5.7179487 7.1538462 6.9743590We are 95% confident that the difference in the mean number calories is between 4.07 and 10.79.
Import the data from Flight Delays Case Study in Section 1.1 data into
R. Although the data are on all UA and AA flights flown in May and June of 2009, we will assume these represent a sample from a larger population of UA and AA flights flown under similar circumstances. We will consider the ratio of the means of the flight delay lengths, \(\mu_{\text{UA}} / \mu_{\text{AA}}\).Perform some exploratory data analysis on flight delay lengths for each of UA and AA flights.
Bootstrap the mean of flight delay lengths for each airline seperately and describe the distribution.
Bootstrap the ratio of means. Provide plots of the bootstrap distribution and describe the distribution.
Find the 95% bootstrap percentile interval for the ratio of means. Interpret this interval.
What is the bootstrap estimate of the bias? What fraction of the bootstrap standard error does it represent?
For inference in this text, we assume that the observations are independent. Is that condition met here? Explain.
FlightDelays <- read.csv("http://www1.appstate.edu/~arnholta/Data/FlightDelays.csv")Your answers:
# Your code here
delays <- subset(FlightDelays, select=c(Carrier, Delay))
p1 <- ggplot(data = delays, aes(x = Delay)) +
geom_histogram(aes(y = ..density..)) +
facet_grid(Carrier ~ .) +
theme_bw()
p2 <- ggplot(data = delays, aes(sample = Delay)) +
stat_qq() +
facet_grid(Carrier ~ .) +
theme_bw()
gridExtra::grid.arrange(p1, p2, ncol = 2)
# Your code here
UA_delays <- delays[delays$Carrier=="UA",]$Delay
AA_delays <- delays[delays$Carrier=="AA",]$Delay
sims <- 10^4
UAboot <- numeric(sims)
AAboot <- numeric(sims)
for (i in 1:sims){
x.UA <- sample(UA_delays, length(UA_delays), replace = TRUE)
x.AA <- sample(AA_delays, length(AA_delays), replace = TRUE)
UAboot[i] <- mean(x.UA)
AAboot[i] <- mean(x.AA)
}
graph1 <- ggplot(as.data.frame(UAboot),aes(x=UAboot)) + geom_histogram(aes(y = ..density..))+ geom_density(size = 1.5)
graph1
graph2 <- ggplot(as.data.frame(AAboot),aes(x=AAboot)) + geom_histogram(aes(y = ..density..))+ geom_density(size = 1.5)
graph2
mean(UAboot)[1] 15.98213sd(UAboot)[1] 1.352021mean(UAboot) - mean(UA_delays)[1] -0.0009492431mean(AAboot)[1] 10.1076sd(AAboot)[1] 0.7452002mean(AAboot) - mean(AA_delays)[1] 0.01021366mean.ratio <- mean(UAboot)/mean(AAboot)The shape of both graphs is a bell curve. The UA boostrap mean is 15.98, the standard error is 1.36, and the bias is -0.0002. The AA bootstrap mean is 10.10, the standard error is 0.74, and the bias is 0.0026. The bias for both of these is small which means we are successful in estimating the lengths of delays.
# Your code here
sims <- 10^4
boot.ratio <- numeric(sims)
for (i in 1:sims){
x.UA <- sample(UA_delays, length(UA_delays), replace = TRUE)
x.AA <- sample(AA_delays, length(AA_delays), replace = TRUE)
boot.ratio[i] <- mean(x.UA)/mean(x.AA)
}
graph1 <- ggplot(as.data.frame(boot.ratio),aes(x=boot.ratio))
graph1 + geom_histogram(aes(y = ..density..))+ geom_density(size = 1.5)
graph2 <- ggplot(as.data.frame(boot.ratio),aes(sample=boot.ratio))
graph2 + stat_qq()
mean(boot.ratio)[1] 1.591342sd(boot.ratio)[1] 0.1786619This graph of the bootstapped ratio is normal. The mean is 1.59 and the standard error is 0.18.
# Your code here
quantileDelay <- quantile(boot.ratio, prob = c(0.025, 0.975))
quantileDelay 2.5% 97.5%
1.263744 1.963118 We are 95% sure that the mean of flight delays bootstrap ratio is between 1.27 and 1.96. This means that we are 95% sure that true mean delay for AA is between 1.27 and 1.96 times less than that for UA.
# Your code here
mean(mean.ratio) - mean(boot.ratio)[1] -0.01014273abs((mean(mean.ratio)-mean(boot.ratio))/sd(boot.ratio)*100)[1] 5.677052The bootstrap distribution has a bias of -0.016. This is 9.29% of the standard error.
- By looking at the data set, you can see that AA and UA have different time tables for arrivals and departures. This would mean the observations are independent.
Two college students collected data on the price of hardcover textbooks from two disciplinary areas: Mathematics and the Natural Sciences, and the Social Sciences (Hien and Baker (2010)). The data are in the file
BookPrices.Perform some exploratory data analysis on book prices for each of the two disciplinary areas.
Bootstrap the mean of the book price for each area separately and describe the distributions.
Bootstrap the ratio of means. Provide plots of the bootstrap distribution and comment.
Find the 95% bootstrap percentile interval for the ratio of means. Interpret this interval.
What is the bootstrap estimate of the bias? What fraction of the bootstrap standard error does it represent?
BookPrices <- read.csv("http://www1.appstate.edu/~arnholta/Data/BookPrices.csv")Your answers:
# Your code here# Your code here# Your code here# Your code here# Your code here