The Empirical Rule The following data represent the

weights $($in grams$)$ of a random sample of $50$ M&M plain

candies.

$0\mathrm{.}870\mathrm{.}880\mathrm{.}820\mathrm{.}900\mathrm{.}900\mathrm{.}840\mathrm{.}84$

$0\mathrm{.}910\mathrm{.}940\mathrm{.}860\mathrm{.}860\mathrm{.}860\mathrm{.}880\mathrm{.}87$

$0\mathrm{.}890\mathrm{.}910\mathrm{.}860\mathrm{.}870\mathrm{.}930\mathrm{.}88$

$0\mathrm{.}830\mathrm{.}950\mathrm{.}870\mathrm{.}930\mathrm{.}910\mathrm{.}85$

$0\mathrm{.}910\mathrm{.}910\mathrm{.}860\mathrm{.}890\mathrm{.}870\mathrm{.}84$

$0\mathrm{.}880\mathrm{.}880\mathrm{.}890\mathrm{.}790\mathrm{.}820\mathrm{.}83$

$0\mathrm{.}900\mathrm{.}880\mathrm{.}840\mathrm{.}930\mathrm{.}810\mathrm{.}90$

$0\mathrm{.}880\mathrm{.}920\mathrm{.}850\mathrm{.}840\mathrm{.}840\mathrm{.}86$

Source: Michael Sullivan.

$($a$)$ Determine the sample standard deviation weight. Express

your answer rounded to three decimal places.

$($b$)$ On the basis of the histogram drawn in Section $3\mathrm{.}1,$

Problem $25,$ comment on the appropriateness of using

the Empirical Rule to make any general statements about

the weights of M&Ms$.$

$($c$)$ Use the Empirical Rule to determine the percentage of

M&Ms with weights between $0\mathrm{.}803$ and $0\mathrm{.}947$ gram. Hint:

x$=0\mathrm{.}875.$

$($d$)$ Determine the actual percentage of M&Ms that weigh

between $0\mathrm{.}803$ and $0\mathrm{.}947$ gram, inclusive.

$($e$)$ Use the Empirical Rule to determine the percentage of

M&Ms with weights more than $0\mathrm{.}911$ gram.