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 $\hat\delta$,
$$ \delta \in \left[ \hat\delta + t_{\alpha/2,2(n-1)} s_p \sqrt{2/n}, \hat\delta + t_{1-\alpha/2,2(n-1)} s_p \sqrt{2/n} \right]. $$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]
## [1] 126.8757 276.4576
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: