Q$4.$ IMARC Group Company has operations in manufacturing eco$-$friendly liquid detergent and

distributed to various supermarkets under a private label brand weekly. There is a production line

designed for filling of $500ml$ bottles. Once these bottles are filled, they are automatically arranged

into packing cases, with each case containing $20$ bottles. In the course of the filling process, the

company established an approximate normal distribution and the standard deviation of the filling

operation as $8ml.$

a$.$ Construct a $95\%$ confidence interval for the mean weight of liquid detergent bottles in a

case.

b$.$ About how many cases would have to be selected such that you would be within $2ml$ of

the population mean value with $99\%$ confidence?

c$.$ The owner of a supermarket in Boston wants to estimate the value of liquid detergents he

has in his store. He takes a random sample of $28$ bottles and determines that the average

retail price of these liquid detergent bottles is $$4\mathrm{.}57$ with a sample standard deviation of $53$

cents. There are $12$ shelves of detergents and the owner estimates that there are $45$ bottles

per shelf. What is the correct value of the standard error of the sample?