Question

5. Fill level of a juice bottle machine is normally distributed with a mean of 12.01 ounces and a standard deviation of 0.1 ounce. a. Find the probability that an individual bottle is filled less than 12.00 ounces. b. Find the probability that a random sample of 24 bottles has a mean fill rate less than 12.00 ounces.

Answer 1

Answer:

a) 46.02% probability that an individual bottle is filled less than 12.00 ounces.

b) 30.85% probability that a random sample of 24 bottles has a mean fill rate less than 12.00 ounces.

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 can be 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 $\frac{\sigma}{\sqrt{n}}$.

In this problem, we have that:

$\mu = 12.01, \sigma = 0.1$

a. Find the probability that an individual bottle is filled less than 12.00 ounces.

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

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

$Z = \frac{12 - 12.01}{0.1}$

$Z = -0.1$

$Z = -0.1$ has a pvalue of 0.4602.

So there is a 46.02% probability that an individual bottle is filled less than 12.00 ounces.

b. Find the probability that a random sample of 24 bottles has a mean fill rate less than 12.00 ounces.

Now we have that $n = 24, s = \frac{0.1}{\sqrt{24}} = 0.02$

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

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

By the Central Limit Theorem, we replace $\sigma$ by s. So

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

$Z = \frac{12 - 12.01}{0.02}$

$Z = -0.5$

$Z = -0.5$ has a pvalue of 0.3085.

So there is a 30.85% probability that a random sample of 24 bottles has a mean fill rate less than 12.00 ounces.