The following data is representative of that reported in an article with x $=$ burner$-$area liberation rate $($MBtu$/$hr$-$ft$2)$ and y $=$ NOx emission rate $($ppm$)$:

x $100125125150150200200250250300300350400400$

y $160150190200190320290390430440400600600680$

$($a$)$ Does the simple linear regression model specify a useful relationship between the two rates? Use the appropriate test procedure to obtain information about the P$-$value, and then reach a conclusion at significance level $0\mathrm{.}01.$

State the appropriate null and alternative hypotheses.

H$0$: $1=0$

Ha: $1>0$ H$0$: $1=0$

Ha: $1<0$ H$0$: $1=0$

Ha: $1!=0$ H$0$: $1!=0$

Ha: $1=0$

Calculate the test statistic and determine the P$-$value. $($Round your test statistic to two decimal places and your P$-$value to three decimal places.$)$

t $=$

P$-$value $=$

State the conclusion in the problem context.

Fail to reject H$0.$ There is no evidence that the model is useful. Reject H$0.$ There is evidence that the model is useful. Reject H$0.$ There is no evidence that the model is useful. Fail to reject H$0.$ There is evidence that the model is useful.

$($b$)$ Compute a $95\%$ CI for the expected change in emission rate associated with a $10$ MBtu$/$hr$-$ft$2$ increase in liberation rate. $($Round your answers to two decimal places.$)$

$,$

ppm