 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 12.11 ounces and a standard deviation of 0.05 ounce. Find the probability that the bottle contains between 12.01 and 12.15 ounces.

Question 8 options:

0.8351

0.8475

0.7654

0.1649

To find the probability that a bottle contains between 12.01 and 12.15 ounces, we will again use the properties of the normal distribution. Given:

- Mean (\(\mu\)) = 12.11 ounces
- Standard deviation (\(\sigma\)) = 0.05 ounces

We need to find the probability \( P(12.01 \leq X \leq 12.15) \).

1. Convert the values 12.01 and 12.15 to their corresponding z-scores using the formula:

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

For \(X = 12.01\):

\[
z_1 = \frac{12.01 - 12.11}{0.05} = \frac{-0.10}{0.05} = -2
\]

For \(X = 12.15\):

\[
z_2 = \frac{12.15 - 12.11}{0.05} = \frac{0.04}{0.05} = 0.8
\]

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

For \(z = -2\):

\[
P(Z \leq -2) = 0.0228
\]

For \(z = 0.8\):

\[
P(Z \leq 0.8) = 0.7881
\]

3. The probability that the bottle contains between 12.01 and 12.15 ounces is the difference between these probabilities:

\[
P(12.01 \leq X \leq 12.15) = P(Z \leq 0.8) - P(Z \leq -2) = 0.7881 - 0.0228 = 0.7653
\]

Therefore, the correct answer is:

\[
\boxed{0.7654}
\]

Would you like more details or have any questions?

Here are 5 numbered questions for you to ask:
1. How do you interpret the z-score in a normal distribution?
2. What is the role of the standard deviation in the normal distribution?
3. How do you find probabilities for values outside the given range in a normal distribution?
4. How can you apply the normal distribution to quality control in manufacturing?
5. What are the steps to find probabilities for any normal distribution problem?

Tip: Always ensure to use the standard normal distribution table or appropriate statistical software for precise probability calculations.

Learn how to calculate the probability that a beer bottle filled by a machine contains between 12.01 and 12.15 ounces using the normal distribution. This problem involves finding probabilities using z-scores, mean, and standard deviation of a normal distribution. Suitable for advanced high school students and anyone interested in statistical probability calculations.

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