Question

A bottling machine can be regulated so that it discharges an average of  ounces
per bottle. It has been observed that the amount of ll dispensed by the machine is
normally distributed with  = 1:0 ounce. A sample of n = 9, lled bottles is randomly
selected from the output of the machine on a given day (all bottled with the same
machine setting), and the ounces of lled bottles are measured. Find the probability
that the sample mean will be within 0:3 ounce of the true mean  for the chosen
machine setting.

Answer 1

The amount of fill dispensed by a bottling machine is normally distributed with $\sigma=1$ ounces and n=9. Suppose $\overline{y}$ be the random variables.

To find : $P(\left | \overline{y}-\mu|\leq 0.3 \right )$

Solution :

$P(\left | \overline{y}-\mu |\leq 0.3 \right)=P(-0.3\leq \overline{y}-\mu\leq 0.3) =P(\frac{-0.3}{\frac{1}{\sqrt{9}}}\leq \frac{\overline{y}-\mu}{\frac{1}{\sqrt{9}}}\leq\frac{0.3}{\frac{1}{\sqrt{9}}})$

$=P(-0.9\leq Z\leq 0.9)$

=0.6318(using z table)

answer: 0.6318