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 12.27 ounces and a standard deviation of 0.06 ounce. Find the probability that the bottle contains between 12.17 and 12.23 ounces.

0.8351

0.20.47

0.8475

0.1649

To find the probability that a bottle contains between 12.17 and 12.23 ounces, we will use the properties of the normal distribution. Given:

- Mean (\(\mu\)) = 12.27 ounces
- Standard deviation (\(\sigma\)) = 0.06 ounces

We need to find the probability \( P(12.17 \leq X \leq 12.23) \).

1. Convert the values 12.17 and 12.23 to their corresponding z-scores using the formula:

\[
z = \frac{X - \mu}{\sigma}
\]

For \(X = 12.17\):

\[
z_1 = \frac{12.17 - 12.27}{0.06} = \frac{-0.10}{0.06} = -1.67
\]

For \(X = 12.23\):

\[
z_2 = \frac{12.23 - 12.27}{0.06} = \frac{-0.04}{0.06} = -0.67
\]

2. Use the standard normal distribution table (or a calculator) to find the probabilities corresponding to these z-scores.

For \(z = -1.67\):

\[
P(Z \leq -1.67) = 0.0475
\]

For \(z = -0.67\):

\[
P(Z \leq -0.67) = 0.2514
\]

3. The probability that the bottle contains between 12.17 and 12.23 ounces is the difference between these probabilities:

\[
P(12.17 \leq X \leq 12.23) = P(Z \leq -0.67) - P(Z \leq -1.67) = 0.2514 - 0.0475 = 0.2039
\]

Therefore, the correct answer is:

\[
\boxed{0.2039}
\]

Would you like more details or have any questions?

Here are 5 numbered questions for you to ask:
1. How do you find z-scores for a given data set?
2. What is the significance of the standard normal distribution table?
3. How do you calculate probabilities for non-standard normal distributions?
4. What are some real-world applications of normal distribution?
5. How does changing the mean or standard deviation affect the normal distribution?

Tip: When dealing with normal distributions, always remember to convert raw scores to z-scores to utilize the standard normal distribution table effectively.

Learn how to calculate the probability that a beer bottle filled by a machine with a mean of 12.27 ounces and standard deviation of 0.06 ounces contains between 12.17 and 12.23 ounces. This problem involves using normal distribution properties and z-scores. Suitable for college-level students.

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