ex0112<-read.csv('ex0112.csv')
fishoil.diet <- ex0112[ex0112$Diet=='FishOil',]
regularoil.diet <- ex0112[ex0112$Diet=='RegularOil',]

Part(a)

n1 <- nrow(fishoil.diet)
n2 <- nrow(regularoil.diet)
mu1 <- mean(fishoil.diet$BP)
mu2 <- mean(regularoil.diet$BP)
s1 <- sd(fishoil.diet$BP)
s2 <- sd(regularoil.diet$BP)

Average of group with diet ‘FishOil’: \(\mu_f=\) 6.5714286

Standard deviation of group with diet ‘FishOil’: \(\sigma_f=\) 5.8554004

Average of group with diet ‘RegularOil’: \(\mu_r=\) -1.1428571

Standard deviation of group with diet ‘RegularOil’: \(\sigma_r=\) 3.1847853

Part(b)

sp <- sqrt( ( (n1-1)*s1^2 + (n2-1)*s2^2 ) / (n1+n2-2) )

Pooled standard deviation = \(s_P=\sqrt{\frac{(n_1-1)s_1^2+(n_2-1)s_2^2}{(n_1+n_2-2)}}\)

\(s_P=\) 4.7132033

Part (c)

se <- sp*sqrt(1/n1+1/n2)

Standard error \(SE(\bar{Y_2}-\bar{Y_1})=\) 2.5193132

Part (d)

df <- n1+n2-2
qt975 <- qt(c(.975), df=df)

Degrees of freedom = \(n_1+n_2-2\)= 12

\(97.5^{th}\) percentile of \(t-distribution\) (\(df=\) 12): 2.1788128

Part (e)

alpha <- 0.05
t <- qt(1-alpha/2,df)
CI_l <- (mu2-mu1)-t*se
CI_h <- (mu2-mu1)+t*se

\(95\%\) CI for \(\mu_2-\mu_1\): \([\)-13.2033975,-2.2251739\(]\) \(t=\) 2.1788128

Part (f)

T <- (mu2-mu1)/se
p <-  pt(T, df=df)

The t-statistic is given by: \(t=\) -3.0620591 with \(df=\) 12

Part (g)

The appropriate one sided p-value is(since \(t<0\)): 0.9975347

Problem (14)

ttest <- t.test( regularoil.diet$BP, fishoil.diet$BP, alternative="greater", var.equal=F)
ttest
## 
##  Welch Two Sample t-test
## 
## data:  regularoil.diet$BP and fishoil.diet$BP
## t = -3.0621, df = 9.2643, p-value = 0.9935
## alternative hypothesis: true difference in means is greater than 0
## 95 percent confidence interval:
##  -12.31752       Inf
## sample estimates:
## mean of x mean of y 
## -1.142857  6.571429

p-value from ‘t.test’ = 0.993458

Problem 19

Part (19a)

n2 <- nrow(regularoil.diet)
mu2 <- mean(regularoil.diet$BP)
s2 <- sd(regularoil.diet$BP)
df2 <- n2-1

Average=: -1.1428571 Standard Devaiation: 3.1847853 Degree of Freedom: 6

Part (19b)

se2 <- s2/sqrt(n2)

Standard error of the average: 1.2037357

Part (19c)

qt975_2 <- qt(c(.975), df=df2)
CI_l2 <- mu2 - qt975_2*se2
CI_h2 <- mu2 + qt975_2*se2

CI: \([\)-4.0882922,1.802578\(]\)

Part 19(d)

T2 <- mu2/se2
p2 <- pt(T2, df=df2)

p-value: 0.1895308

t-statistics: -0.9494253