A manufacturer produces both a deluxe and a standard model of an automatic sander designed for home use. Selling prices obtained from a sample of retail outlets follow.

Retail

Outlet Model Price $($$$)$

Deluxe Standard

$13927$

$23929$

$34536$

$43830$

$54030$

$63934$

$73529$

$($a$)$

The manufacturer's suggested retail prices for the two models show a $$10$ price differential. Use a $0\mathrm{.}05$ level of significance and test that the mean difference between the prices of the two models is $$10.$

State the null and alternative hypotheses. $($Use $$d $=$ mean price for the deluxe model $$ mean price for the standard model.$)$

Calculate the value of the test statistic. $($Round your answer to three decimal places.$)$

Calculate the p$-$value. $($Round your answer to four decimal places.$)$

p$-$value $=$

State your conclusion.

Reject H$0.$ There is sufficient evidence to conclude that a $$10$ price differential exists.

Do not reject H$0.$ There is insufficient evidence to conclude that a $$10$ price differential exists.

Do not Reject H$0.$ There is sufficient evidence to conclude that a $$10$ price differential exists.

Reject H$0.$ There is insufficient evidence to conclude that a $$10$ price differential exists.

$($b$)$

What is the $95\%$ confidence interval for the difference between the mean prices of the two models $($in dollars$)?($Round your answers to nearest cent. Use the mean price for the deluxe model $$ the mean price for the standard model.$)$