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.