Q.9 - The Practice of Statistics for AP

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  

Step-by-Step Solution

Answer

The correct answer is (e)

Step 1 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 $=4$

Step 2 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).