Question

Bottles filled by a certain machine are supposed to contain 12 oz of liquid. In fact the fill volume is random with mean 12.01 oz and standard deviation 0.20 oz.What is the probability that the mean volume of a random sample of 144 bottles is less than 12 oz

Answer 1

Answer:

27.43% probability that the mean volume of a random sample of 144 bottles is less than 12 oz.

Step-by-step explanation:

To solve this problem, it is important to understand the normal probability distribution and the central limit theorem.

Normal probability distribution

Problems of normally distributed samples are solved using the z-score formula.

In a set with mean $\mu$ and standard deviation $\sigma$, the zscore of a measure X is given by:

$Z = \frac{X - \mu}{\sigma}$

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

Central Limit Theorem

The Central Limit Theorem estabilishes that, for a random variable X, with mean $\mu$ and standard deviation $\sigma$, a large sample size can be approximated to a normal distribution with mean $\mu$ and standard deviation $s = \frac{\sigma}{\sqrt{n}}$

In this problem, we have that:

$\mu = 12.01, \sigma = 0.2, n = 144, s = \frac{0.2}{\sqrt{144}} = 0.0167$

What is the probability that the mean volume of a random sample of 144 bottles is less than 12 oz

This is the pvalue of Z when $X = 12$

$Z = \frac{X - \mu}{\sigma}$

By the Central Limit Theorem

$Z = \frac{X - \mu}{s}$

$Z = \frac{12 - 12.01}{0.0167}$

$Z = -0.6$

$Z = -0.6$ has a pvalue of 0.2743.

So there is a 27.43% probability that the mean volume of a random sample of 144 bottles is less than 12 oz.