Question

A machine is designed to fill $16$-ounce bottles of shampoo. When the machine is working properly, the mean amount poured into the bottles is $16.05$ounces with a standard deviation of $0.1$ounces. Assume that the machine is working properly. If four bottles are randomly selected each hour and the number of ounces in each bottle is measured, then $95\%$of the observations should occur in which interval?

(a) $16.05$to $16.15$ounces

(b) $0.30$to $0.30$ounces

(c) $15.95$to$16.15$ ounces

(d) $15.90$to $16.20$ounces

(e) None of the above

Short answer

The correct answer is (e)

Step-by-step solution

Given Information

Amount of bottles of shampoo to fill the machıne $=16$ounces

Mean amount of shampoo $=16.05$ounces

Standard deviation $=0.1$ounce

Number of randomly selected bottles$=4$

Explanation

Consider that $n$represents the number of randomly selected bottles that is, $n=4;\sigma$represents the standard deviation that is $\sigma =0.1$, and $\mu$denotes the mean that is, $\mu =16.05$.

In the case of a Normal distribution, it is known that $95\%$the observations lie within $2$standard deviations of the mean. But it is not given that the population distribution is normal so the central limit theorem cannot be applied.

Hence, the correct answer is (e).