Problem Set 1

Ethan Witkowski Spring 2019

3a)

Define Variables:

xbar <- 121.07
mu <- 120
SD <- 16.59
n = 371
alpha = .05
SDx = (SD/sqrt(n))
df = n-1

tstat <- (xbar-mu)/SDx
tstat

## [1] 1.242294

pvalue <- 2*pt(-abs(tstat),df)
pvalue

## [1] 0.2149149

Because >.05, we fail to reject the null, and these control subjects do not differ significantly from a normal population.

3b)

n2 <- 3710
SDx2 <- (SD/sqrt(n2))
df2 <- 3709

tstat_b <- (xbar-mu)/SDx2
tstat_b

## [1] 3.928478

pvalue_b <- 2*pt(-abs(tstat_b),df2)
pvalue_b

## [1] 8.704141e-05

Effect size is the magnitude of difference between means; statistical significance is our confidence that the effect size is not occurring due to chance.

An increase in n, holding the effect size constant, increases the statistical significance.

3c)

CIlower <- (mu-(1.96*SDx2))
CIlower

## [1] 119.4662

CIhigher <- (mu+(1.96*SDx2))
CIhigher

## [1] 120.5338

This confidence interval does contain 120.

4a)

Generate random samples, create 4 histograms.

par(mfrow=c(2,2))
for (i in 1:4) {
  dataset <- rnorm(10,0,1)
  hist(dataset)
}

The samples do not appear to be normal.

4b)

Generate random sample, create 4 QQplots.

par(mfrow=c(2,2))
for (i in 1:4) {
  dataset <- rnorm(10,0,1)
  qqnorm(dataset)
  abline(0,1,col=gray(.7))
}

The samples do not appear to be normal.

4c)

Repeat sampling with 100 observations.

par(mfrow=c(2,2))
for (i in 1:4) {
  dataset <- rnorm(100,0,1)
  hist(dataset)
}

par(mfrow=c(2,2))
for (i in 1:4) {
  dataset <- rnorm(100,0,1)
  qqnorm(dataset)
  abline(0,1,col=gray(.7))
}

The samples appear to more closely resemble a normal distribution.

5a)

datasetbmi <- read.csv("C:\\Users\\ethan\\Desktop\\Swarthmore\\Spring 2019\\Statistics II\\Problem Sets\\Problem Set 1\\bmi.csv", header=T)

bmi <- datasetbmi[,"bmi"]
status <- datasetbmi[,"status"]

5b)

meandiff <- mean(bmi[status==1]) - mean(bmi[status==99])
meandiff

## [1] 3.431874

5c)

numshuffles <- 1000
diffs <- rep(NA, numshuffles)

for(i in 1:numshuffles) {
  newstatus <- sample(status)
  diffs[i] <- mean(bmi[newstatus==1]) - mean(bmi[newstatus==99])
}

5d)

par(mfrow=c(1,1))
hist(diffs)
abline(v=meandiff, col="red")

pvalue_diff <- meandiff/numshuffles
pvalue_diff

## [1] 0.003431874

The null hypothesis is rejected because <.05.

6a)

marathon <- scan("C:\\Users\\ethan\\Desktop\\Swarthmore\\Spring 2019\\Statistics II\\Problem Sets\\Problem Set 1\\nycmarathon.csv")

par(mfrow=c(2,1))

hist(marathon, col="darkgrey", border="white", nclass=200)
hist(marathon, col="darkgrey", border="white", nclass=30)