Problem 1

An experiment is conducted to study the effect of drilling method on drilling time. Each method (dry drilling, wet drilling) is used on $n = 12$ rocks. Drilling times are measured in $1/100$ minutes.

Part (a)

\fbox{\begin{minipage}{\textwidth} Compute a $95$ confidence interval for $\delta = \mu_1 - \mu_2$. Provide an interpretation, stated in the context of the problem. \end{minipage}}

A confidence interval for $\delta$ includes all parameter values compatible with the observed data $\hat\delta$,

We compute this CI with:

library("readxl")
data = read_excel("./handout1data.xlsx")
data$method = as.factor(data$method)
dry = na.omit(data$time[data$method=='d'])
wet = na.omit(data$time[data$method!='d'])
alpha = .05

t.test(x=dry,
       y=wet,
       alternative=c("two.sided"),
       conf.level=1-alpha,
       var.equal=T)$conf.int[1:2]
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## [1] 126.8757 276.4576
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We estimate that the difference in drilling methods, $\delta = (\rm{dry} - \rm{wet})$, is between $[126.876,276.458]$.

Part (b)

\fbox{\begin{minipage}{\textwidth} Explain how a confidence interval provides a complementary result to a hypothesis test. \end{minipage}}

A hypothesis test looks to determine if an effect exists. A CI looks to determine the size of the effect.

Problem 2

A prodcut developer is investigating the tensile strength of a new synthetic fiber. A completely randomized design with five levels of cotton content is performed, with $n=5$ speciments per level.

Part (a)

\fbox{\begin{minipage}{\textwidth} Compute and display $95$ confidence intervals for all pairwise comparisons. \end{minipage}}

We show the confidence intervals with: