Bluesea Seafoods are a supplier of pre$-$packed seafood to the two major supermarket chains. Their Quality Assurance Officer is responsible for ensuring the pre$-$packaged items comply with their marked weights. They routinely inspect their smoked salmon product to determine if the contents of these packets weight at least the $180$ grams advertised. A random sample of $55$ packets of smoked salmon, whose advertised weight claims to be $180$ grams, yielded the following statistics:

x$=181$ grams

s$=5\mathrm{.}5$ grams

Construct a $95\%$ confidence interval estimate for the mean weight of the pre$-$packaged smoked salmon with an advertised weight of $180$ grams. Hint: clearly justify the choice of the sampling distribution of the sample mean. Include the steps of the confidence interval estimation and appropriate critical values.