The sanitation department of a large city wants to investigate ways to reduce the amount of recyclable materials that are placed in the city$’$s landfill. The city hopes to prolong the life of the landfill site. From an analysis of recycling records from other cities, it is determined that if the average weekly amount of recyclable material is more than $5$ pounds per household, a commercial recycling firm could make a profit collecting the material. To determine the feasibility of the recycling plan, a random sample of $50$ households is selected. The weekly weight of recyclable material $($in pounds$)$ is recorded for each household:

n $=50,$ sample mean$=12\mathrm{.}46,$ sample standard deviation$=11\mathrm{.}75$

$($a$)$ Define the parameter involved and state the null and alternative hypotheses using the parameter.

$($b$)$ Perform the hypothesis test at the $5\%$ level of significance. What is the p$-$value? Interpret the result in the context.