Rutter Nursery Company packages its pine bark mulch in $50-$pound bags. From a long history, management knows that the distribu f bag weights is normally distributed with a population standard deviation of $3$ pounds per bag. At the end of each day, Jeff Rutter, production manager, weighs $10$ bags and computes the mean weight of the sample. Following are the weights of $10$ bags from toda roduction.

$\backslash$table$[[45\mathrm{.}6,47\mathrm{.}7,47\mathrm{.}6,46\mathrm{.}3,46\mathrm{.}2,47\mathrm{.}4,49\mathrm{.}2,55\mathrm{.}8,47\mathrm{.}5,48\mathrm{.}5]]$

Can Mr$.$ Rutter conclude that the mean weight of the bags is less than $50$ pounds? Use the $0\mathrm{.}01$ significance level.

Required:

a$.$ What is the decision rule?

Note: Negative amount should be indicated by a minus sign. Round your answer to $2$ decimal places.

Reject $HO$:$50$ when the test statistic is

b$.$ Compute the value of the test statistic.

Note: Negative amount should be indicated by a minus sign. Round your answer to $2$ decimal places.

Value of the test statistic

c$.$ What is your decision regarding $H_0?$

$H_0$ and conclude that the mean weight

less than $50$ pounds.