Suppose a brewery has a filling machine that fills 12 ounce bottles of beer. It is known that the amount of beer poured by this filling machine follows a normal distribution with a mean of 12.14 onces and a standard deviation of 0.04 ounce. Find the probability that the bottle contains more than 12.14 ounces of beer.
Continuous Probability Distributions
Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.
Normal Distribution
Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!
![**Example Problem**
**Scenario:** Suppose a brewery has a filling machine that fills 12-ounce bottles of beer. It is known that the amount of beer poured by this filling machine follows a normal distribution with a mean of 12.14 ounces and a standard deviation of 0.04 ounce.
**Question:** Find the probability that the bottle contains more than 12.14 ounces of beer.
**Solution:**
1. **Identify the given parameters:**
- Mean (\(\mu\)) = 12.14 ounces
- Standard deviation (\(\sigma\)) = 0.04 ounce
2. **State the problem in terms of probability:**
- We need to find the probability that a bottle contains more than 12.14 ounces of beer. Mathematically, this is \(P(X > 12.14)\), where \(X\) is the amount of beer in a bottle.
3. **Standardize the normal distribution:**
- Convert the value to a Z-score using the formula:
\[
Z = \frac{X - \mu}{\sigma}
\]
For \(X = 12.14\):
\[
Z = \frac{12.14 - 12.14}{0.04} = 0
\]
4. **Find the probability using the standard normal distribution:**
- Using the Z-score table, the probability \(P(Z > 0)\) corresponds to the area to the right of Z = 0.
- From the Z-table, \(P(Z \le 0) = 0.5\).
5. **Determine the final probability:**
- Since the total area under the normal distribution curve is 1:
\[
P(Z > 0) = 1 - P(Z \le 0) = 1 - 0.5 = 0.5
\]
**Answer:** The probability that the bottle contains more than 12.14 ounces of beer is 0.5, or 50%.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffa41b47d-a27d-4623-80de-b3b064f82b52%2Fb71b399c-ed35-499d-b16f-86ea340a4afb%2Ff4bxeum_processed.jpeg&w=3840&q=75)
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