To determine the lengths for which a bolt will be destroyed, we use the given policy: any bolt that is more than 2 standard deviations away from the mean will be destroyed.

Key Information:

• Mean bolt length ($\mu$): 99 cm
• Standard deviation ($\sigma$): 0.10 cm
• The bolt is destroyed if its length is more than 2 standard deviations from the mean.

Step 1: Calculate the range of acceptable lengths

The acceptable range is within 2 standard deviations of the mean. We calculate the lower and upper limits as follows:

• Lower limit: $\mu - 2\sigma$
• Upper limit: $\mu + 2\sigma$

Now, substitute the given values:

$\text{Lower limit} = 99 - 2(0.10) = 99 - 0.20 = 98.80 \, \text{cm}$ $\text{Upper limit} = 99 + 2(0.10) = 99 + 0.20 = 99.20 \, \text{cm}$

Step 2: Interpret the results

Any bolt that is shorter than 98.80 cm or longer than 99.20 cm will be destroyed.

Final Answer:

• Bolts shorter than 98.80 cm or longer than 99.20 cm will be destroyed.

Would you like more details on the standard deviation, or have any questions?

Here are some related questions to consider:

1. What is a standard deviation, and why is it used in quality control?
2. How would the range change if the standard deviation increased?
3. Can you explain how the normal distribution relates to this problem?
4. What would happen if the quality control allowed a deviation of 3 standard deviations instead of 2?
5. How can we calculate the percentage of bolts expected to fall outside this range?

Tip: Always ensure to calculate both lower and upper limits when applying standard deviation to set ranges.