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

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.

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 = (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
`(\sigma)/(√(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 = (X - \mu)/(\sigma)`

`Z = (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 = (0.1)/(√(24)) = 0.02`

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

`Z = (X - \mu)/(\sigma)`

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

`Z = (X - \mu)/(s)`

`Z = (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.