Let's analyze the given problem step by step.

We are dealing with a normal distribution, where:

• The mean value of the filling volume is $\mu = 331$ ml.
• The standard deviation is $\sigma = 1.34$ ml.

The task defines a "gravierende Minderbefüllung" (significant underfilling) as any bottle filled with $\leq 327$ ml.

Task (1)

Determine the probability that a randomly chosen bottle has a significant underfilling.

To solve this, we need to find: $P(Y \leq 327)$ Where $Y \sim N(\mu = 331, \sigma = 1.34)$.

To calculate this, we use the Z-score formula: $Z = \frac{X - \mu}{\sigma}$ Where:

• $X = 327$
• $\mu = 331$
• $\sigma = 1.34$

Let me calculate this probability.It seems there was an issue with calculating the result. I’ll manually walk through the solution.

We use the Z-score formula to standardize the problem:

$Z = \frac{327 - 331}{1.34} = \frac{-4}{1.34} \approx -2.9851$

Now, using the cumulative distribution function (CDF) for a normal distribution, we find $P(Z \leq -2.9851)$.

Using standard normal distribution tables or a calculator, we get:

$P(Z \leq -2.9851) \approx 0.0014$

Thus, the probability that a randomly chosen bottle has a significant underfilling is approximately 0.0014.

Let me know if you have any questions or would like further details! Here are 5 related questions:

1. How would the probability change if the mean filling were reduced to 330 ml?
2. What is the Z-score for a filling of 328 ml?
3. How does increasing the standard deviation affect the probability of underfilling?
4. What would the probability of underfilling be if the threshold were set at 326 ml?
5. How can we use this model to assess the quality control process in production?

Tip: For small probabilities in normal distributions, Z-tables or cumulative functions are essential for quick calculations.