erform the test in problem S$2$ using a bootstrap analysis.

$($a$)$ Would we want to use a one$-$sample or two$-$sample bootstrap test? Explain your choice.

$($b$)$ The code below runs the bootstrap procedure on the differences between the shoot and root

data. However, there are some problems with the code.

$$ The code only runs the bootstrap procedure $50$ times, which is not enough to make an

accurate approximation of the null distribution.

$$ The code takes a resample without replacement.

Fix the code so that the bootstrap procedure runs $5000$ times and samples with

replacement.

Shoot $<-$ c$(4\mathrm{.}42,5\mathrm{.}81,4\mathrm{.}65,4\mathrm{.}77,5\mathrm{.}25,4\mathrm{.}75)$

Root $<-$ c$(3\mathrm{.}76,5\mathrm{.}40,3\mathrm{.}91,4\mathrm{.}29,4\mathrm{.}69,3\mathrm{.}93)$

diff $<-$ Shoot $-$ Root

# Calculate summary statistics

x$\_$bar $<-$ mean$($diff$)$

n $<-$ length$($diff$)$

B $<-50$

# Create a vector to store t$\_$hat values

t$\_$hat $<-$ numeric$($B$)$

set.seed$(371)$

# Bootstrap loop

for$($i in $1$:B$)\{$

# $2.$ Draw a SRS of size n from data

x$\_$star $<-$ sample$($diff$,$ size $=$ n$,$ replace $=$ F$)$

# $3.$ Calculate resampled mean and sd

x$\_$bar$\_$star $<-$ mean$($x$\_$star$)$

s$\_$star $<-$ sd$($x$\_$star$)$

$6$

# $4.$ Calculate t$\_$hat, and store it in vector

t$\_$hat$[$i$]<-($x$\_$bar$\_$star $-$ x$\_$bar$)/($s$\_$star$/$sqrt$($n$))$

$\}$

$($c$)$ Calculate a two$-$sided bootstrap p$-$value.

$($d$)$ Explain why the bootstrap test result is similar to the T test result from S$2.$