Chapter 7: Q. 57 (page 479)

Bottling cola A bottling company uses a filling machine to fill plastic bottles with cola. The bottles are supposed to contain 300 milliliters $(ml)$ . In fact, the contents vary according to a Normal distribution with mean $μ = 298 ml$ and standard deviation $σ = 3 ml$ .
a. What is the probability that a randomly selected bottle contains less than $295 ml$ ?
b. What is the probability that the mean contents of six randomly selected bottles is less than $295 ml$ ?

Short Answer

Expert verified

(a). the probability that a randomly selected bottle contains less than $295 m l$ 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,

μ = 298 σ = 3 x = 295

Use the following formulae

$z = \frac{x - μ}{σ}$

Part(a) Step 2: Calculation

The $z$ -score is

$z = \frac{x - μ}{σ} = \frac{295 - 298}{3} = - 1.00$

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,

μ = 298 σ = 3 n = 6 x = 295

Part(b) Step 2: Calculation

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

The $z$ -score is

$z = \frac{x - μ_{\bar{\bar{x}}}}{σ_{\bar{\bar{x}}}} = \frac{\bar{x} - μ}{d \sqrt{n}} = \frac{295 - 298}{3 \sqrt{\bar{n}}} = - 2.45$

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