A manufacturer of bolts has a​ quality-control policy that requires it to destroy any bolts that are more than 22 standard deviations from the mean. The​ quality-control engineer knows that the bolts coming off the assembly line have mean length of 99 cm with a standard deviation of 0.100.10 cm. For what lengths will a bolt be​ destroyed?

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.

A manufacturer of bolts has a​ quality-control policy that requires it to destroy any bolts that are more than 2 standard deviations from the mean. The​ quality-control engineer knows that the bolts coming off the assembly line have mean length of 99 cm with a standard deviation of 0.10 cm. For what lengths will a bolt be​ destroyed?

Learn how to determine the acceptable length range for bolts based on a manufacturer’s quality-control policy. Using a mean length of 99 cm and a standard deviation of 0.10 cm, find out which bolts will be destroyed if they deviate more than 2 standard deviations from the mean. Suitable for students in Grades 9-12.

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