$20.$ Assume that two samples are independent simple random samples selected from normally distributed populations. Do not assume that the population standard deviations are equal. A baker wished to compare the baking times of two different types of pastries. Independent simple random samples of $25$ brownies and $28$ croissants were observed. The baking times, in minutes, were recorded, and the summary statistics are below. Construct a $90\%$ confidence interval for $\backslash$mu $1-\backslash$mu $2,$ the difference between the mean baking time for croissants and the mean baking time for brownies.

Brownies Croissants

$1=29$ mins

$2=20$ mins

s$1=1\mathrm{.}2$ mins s$2=1\mathrm{.}7$ mins

n$1=25$ n$2=28$

a$.1\mathrm{.}8<\backslash$mu $1-\backslash$mu $2<19\mathrm{.}8$

b$.9\mathrm{.}5<\backslash$mu $1-\backslash$mu $2<27\mathrm{.}5$

c$.9\mathrm{.}4<\backslash$mu $1-\backslash$mu $2<20\mathrm{.}2$

d$.1\mathrm{.}3<\backslash$mu $1-\backslash$mu $2<22\mathrm{.}9$