A particular brand of diet margarine was analyzed to determine the level of polyunsaturated fatty acid $($in percentages$).$ A sample of six packages resulted in the following data: $16\mathrm{.}8,17\mathrm{.}2,17\mathrm{.}4,16\mathrm{.}9,16\mathrm{.}5,17\mathrm{.}1,17\mathrm{.}9.$

Calculate a $99\%$ confidence interval on the mean $\backslash$mu $.$Provide a practical interpretation of this interval.

Calculate a $99\%$ lower confidence bound on the mean. Compare this bound with the lower bound of the two$-$sided confidence interval and discuss why they are different.

Compute a $99\%$ prediction interval on the polyunsaturated fatty acid in the next package of margarine that is tested.

Compare the length of the prediction interval with the length of the $99\%$ CI on the population mean.