Question

Bottling cola A bottling company uses a filling machine to fill plastic bottles with cola. The bottles are supposed to contain 300 milliliters $\left(\mathrm{ml}\right)$. In fact, the contents vary according to a Normal distribution with mean $\mu =298\mathrm{ml}$and standard deviation $\sigma =3\mathrm{ml}$.

a. What is the probability that a randomly selected bottle contains less than $295\mathrm{ml}$?

b. What is the probability that the mean contents of six randomly selected bottles is less than $295\mathrm{ml}$?

Short answer

(a). the probability that a randomly selected bottle contains less than $295ml$is $15.87\%$

(b). the probability that the mean contents of six randomly selected bottles is less than is$0.71\%$

Step-by-step solution

Part(a) Step 1: Given information 

Given,

$$\begin{gathered} \mu =298 \\ \sigma =3 \\ x=295 \end{gathered}$$

Use the following formulae

$z=\frac{x-\mu }{\sigma }$

Part(a) Step 2: Calculation

The $z$-score is

$$\begin{gathered} z=\frac{x-\mu }{\sigma } \\ =\frac{295-298}{3} \\ =-1.00 \end{gathered}$$

The likelihood of associating using the normal probability $P(Z<-1.00)$is given in the row starting with $-1.0$and in the column starting with $.00$of the standard normal probability

$P(x<295)=P(Z<-1.00)=0.1587=15.87\%$

Part(b) step 1: Given information 

Given,

$$\begin{gathered} \mu =298 \\ \sigma =3 \\ n=6 \\ x=295 \end{gathered}$$

Part(b) Step 2: Calculation

Because the population distribution is normal, so is the sampling distribution of the sample mean $\overline{x}$.

The $z$-score is

$$\begin{gathered} z=\frac{x-{\mu }_{\overline{\overline{x}}}}{{\sigma }_{\overline{\overline{x}}}} \\ =\frac{\overline{x}-\mu }{d\sqrt{n}} \\ =\frac{295-298}{3\sqrt{\overline{n}}} \\ =-2.45 \end{gathered}$$

The associating probability using the normal probability $P(Z<-2.45)$is given in the row starting with $-2.4$and in the column starting with $.05$of the standard normal probability