The amount of juice in bottles labeled as containing 40 ounces is normally distributed with a mean of 41.28 ounces and a standard deviation of 1.39 ounces. What proportion of bottles are filled with less than 40 ounces of juice?
Round your answer to 4 places after the decimal point.

To find the proportion of bottles filled with less than 40 ounces of juice, we can standardize the given value using the **z-score** formula, which represents how many standard deviations a value is away from the mean in a normal distribution.

The formula for the z-score is:

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

Where:
- \( x = 40 \) (the value we are interested in),
- \( \mu = 41.28 \) (the mean of the distribution),
- \( \sigma = 1.39 \) (the standard deviation of the distribution).

### Step 1: Calculate the z-score

\[
z = \frac{40 - 41.28}{1.39} = \frac{-1.28}{1.39} \approx -0.9209
\]

### Step 2: Find the proportion

Next, we need to look up the z-score in a standard normal distribution table, or use a calculator to find the proportion of the distribution that falls below this z-score. For \( z = -0.9209 \), the cumulative probability is approximately 0.1788.

Thus, the proportion of bottles filled with less than 40 ounces is **0.1788**.

### Final Answer:
\[
\text{Proportion} \approx 0.1788
\]

Would you like further details or have any questions? Here are some related problems for practice:

1. What proportion of bottles contain more than 42 ounces of juice?
2. What is the probability that a bottle contains between 40 and 42 ounces?
3. If the standard deviation changes to 1.5, how does the proportion change?
4. What is the z-score for a bottle containing exactly 41 ounces?
5. What amount of juice corresponds to the 90th percentile?

**Tip**: The z-score allows you to compare different distributions by converting them to a standard form, enabling easier comparison.

The amount of juice in bottles labeled as containing 40 ounces is normally distributed with a mean of 41.28 ounces and a standard deviation of 1.39 ounces. What proportion of bottles are filled with less than 40 ounces of juice? Round your answer to 4 places after the decimal point.

Learn how to calculate the proportion of juice bottles filled with less than 40 ounces using the normal distribution and z-score formula. This problem involves statistical concepts such as mean, standard deviation, and cumulative probability, and is suitable for students in Grades 11-12.

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