The weight of 9-ounce bags of this particular brand of potato chips follows a normal distribution with mean $\mu = 9.12$ ounces and standard deviation $\sigma = 0.05$ ounces.

We want to determine what percentage of bags weigh less than the advertised 9 ounces.

Step 1: Calculate the Z-score

The Z-score formula is given by:

$Z = \frac{X - \mu}{\sigma}$

where:

• $X = 9$ ounces (the value we're interested in),
• $\mu = 9.12$ ounces (the mean),
• $\sigma = 0.05$ ounces (the standard deviation).

Substituting the values:

$Z = \frac{9 - 9.12}{0.05} = \frac{-0.12}{0.05} = -2.4$

Step 2: Find the percentage corresponding to the Z-score

A Z-score of $-2.4$ means that we need to find the proportion of the normal distribution that lies below this Z-score. Using Z-tables or a normal distribution calculator, the probability corresponding to a Z-score of $-2.4$ is approximately 0.0082, or 0.82%.

Step 3: Interpretation

About 0.82% of the 9-ounce bags weigh less than the advertised 9 ounces.

Step 4: Is this likely to pose a problem for the company?

Since only about 0.82% of the bags are underweight (less than 9 ounces), this is a small percentage. In practice, this would likely not pose a significant problem for the company, as most bags are above the advertised weight. However, the company would still need to consider consumer perceptions and quality control measures to ensure that a few underweight bags don't affect their brand reputation.

Do you want more details or have any questions?

Here are 5 related questions to expand this topic:

1. How does changing the standard deviation affect the percentage of underweight bags?
2. What percentage of bags weigh more than 9.2 ounces?
3. What is the probability that a bag weighs between 9.05 and 9.15 ounces?
4. How could the company adjust the mean weight to further minimize the chance of underweight bags?
5. How can the company ensure that the variability (standard deviation) in the bag weight remains low?

Tip: To reduce the likelihood of underweight bags, the company could consider adjusting its production process to consistently fill slightly above the target weight.