A factory produces light bulbs, and the lifetime of these bulbs is normally distributed with a known standard deviation σ=100 hours. A sample of 36 light bulbs has a mean lifetime of 1,200 hours. Construct a 95% confidence interval for the true mean lifetime of the bulbs.

To construct a 95% confidence interval for the true mean lifetime of the light bulbs, we will use the formula for a confidence interval for a population mean when the population standard deviation (σ) is known:

\[
CI = \bar{X} \pm Z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}}
\]

Where:
- \(\bar{X} = 1,200\) (sample mean)
- \(σ = 100\) hours (population standard deviation)
- \(n = 36\) (sample size)
- \(Z_{\alpha/2} = 1.96\) (critical value from the standard normal distribution for a 95% confidence interval)

### Step 1: Calculate the standard error (SE) of the mean
\[
SE = \frac{σ}{\sqrt{n}} = \frac{100}{\sqrt{36}} = \frac{100}{6} = 16.67
\]

### Step 2: Calculate the margin of error (ME)
\[
ME = Z_{\alpha/2} \cdot SE = 1.96 \cdot 16.67 = 32.67
\]

### Step 3: Calculate the confidence interval
\[
CI = \bar{X} \pm ME = 1,200 \pm 32.67
\]
\[
CI = (1,200 - 32.67, 1,200 + 32.67) = (1,167.33, 1,232.67)
\]

### Conclusion:
The 95% confidence interval for the true mean lifetime of the light bulbs is **(1,167.33 hours, 1,232.67 hours)**.

Would you like more details or further explanation on any step?

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Here are some related questions that expand on this topic:
1. How does the confidence level affect the width of the confidence interval?
2. What happens if the sample size increases in this context?
3. How would the confidence interval change if the population standard deviation was unknown?
4. Can you explain why a normal distribution is important here?
5. How would a 99% confidence interval differ from the 95% confidence interval?

**Tip:** Increasing the sample size decreases the margin of error, resulting in a narrower confidence interval.

This solution explains how to construct a 95% confidence interval for the true mean lifetime of light bulbs produced by a factory. Given a sample mean of 1,200 hours, a standard deviation of 100 hours, and a sample size of 36, the confidence interval is calculated as (1,167.33, 1,232.67). The solution covers key statistical concepts like the normal distribution and margin of error.

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