To ensure a 99.9% probability that a half-liter (500 ml) soft drink bottle is not underfilled, we need to set the mean of the filling process appropriately. Here's how to calculate it:

Key Information:

• Desired probability that the bottle is not underfilled: 99.9% (which means the probability of underfilling is 0.1%).
• The process has a standard deviation ($\sigma$) of 5 ml.
• The bottle is considered underfilled if it contains less than 500 ml.
• We want to find the mean ($\mu$) that ensures 99.9% of bottles are filled with at least 500 ml.

Step-by-Step Solution:

1. Identify the corresponding z-score: Since we need a 99.9% probability that the bottle is not underfilled, this implies that we are looking for the 0.1st percentile in the lower tail of the normal distribution. The z-score corresponding to the 0.1st percentile (99.9% to the right) is:

   $z = -3.090$

2. Set up the equation: We know the z-score formula for a normal distribution:

   $z = \frac{x - \mu}{\sigma}$

   Where:

   • $x$ is the target fill level (500 ml),
   • $\mu$ is the mean we need to find,
   • $\sigma$ is the standard deviation (5 ml).

   Substituting $z = -3.090$, $x = 500$, and $\sigma = 5$, we get:

   $-3.090 = \frac{500 - \mu}{5}$

3. Solve for the mean ($\mu$):

   $-3.090 \times 5 = 500 - \mu$

   $-15.45 = 500 - \mu$

   $\mu = 500 + 15.45$

   $\mu = 515.45$

Conclusion:

The mean should be set to 515.45 ml to ensure that there is a 99.9% probability that a half-liter bottle will not be underfilled.

Would you like more details or have any questions about this process?

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Here are 5 related questions you might be interested in:

1. How do you adjust the mean if the desired probability is 99% instead of 99.9%?
2. What happens to the required mean if the standard deviation increases?
3. How do you calculate the probability of overfilling the bottle?
4. How does changing the bottle size affect the filling process?
5. How can you reduce the standard deviation in the filling process?

Tip: Always remember that the z-score represents the number of standard deviations away from the mean a certain value is in a normal distribution.