Let's solve the questions using the empirical rule, which helps us approximate the percentage of data that falls within one, two, or three standard deviations from the mean in a normal distribution.

Given:

• Mean ($\mu$) = 9.12 ounces
• Standard deviation ($\sigma$) = 0.05 ounces

Part (a): What percent of bags weigh less than 9.07 ounces?

1. Calculate how many standard deviations 9.07 ounces is below the mean:

$Z = \frac{9.07 - 9.12}{0.05} = \frac{-0.05}{0.05} = -1$

A value of 9.07 is 1 standard deviation below the mean.

Using the empirical rule:

• About 68% of the data falls within 1 standard deviation from the mean (i.e., between 9.07 and 9.17 ounces).
• The remaining 32% is split equally between the tails (below 9.07 and above 9.17).
• Hence, 16% of the bags weigh less than 9.07 ounces.

Answer for (a): 16%

Part (b): What proportion of bags weigh between 8.97 and 9.17 ounces?

1. Calculate how many standard deviations these values are from the mean:

• For 9.17 ounces:

$Z = \frac{9.17 - 9.12}{0.05} = \frac{0.05}{0.05} = 1$

• For 8.97 ounces:

$Z = \frac{8.97 - 9.12}{0.05} = \frac{-0.15}{0.05} = -3$

So, 9.17 is 1 standard deviation above the mean, and 8.97 is 3 standard deviations below the mean.

Using the empirical rule:

• About 99.7% of the data falls within 3 standard deviations.
• About 68% of the data falls within 1 standard deviation (from 9.07 to 9.17 ounces).

Thus, the proportion of bags that weigh between 8.97 and 9.17 ounces would be approximately 99.7% (since this range covers up to 3 standard deviations from the left and 1 standard deviation on the right).

Answer for (b): 99.7%

Let me know if you need any clarifications or further details on the process!

Here are 5 additional related questions you could explore:

1. What percentage of bags weigh more than 9.17 ounces?
2. What percentage of bags weigh less than 8.97 ounces?
3. What percentage of bags weigh between 9.07 and 9.12 ounces?
4. How many standard deviations would a weight of 9.22 ounces be from the mean?
5. If the mean weight shifts to 9.10 ounces, how would that change the percentages?

Tip:

The empirical rule is useful for quick estimates but using a Z-table or normal distribution calculator would give more precise answers!