The lifetime of a certain brand of electric light bulb is known to have a standard deviation of 54 hours. Suppose that a random sample of 50 bulbs of this brand has a mean lifetime of 486 hours. Find a 95% confidence interval for the true mean lifetime of all light bulbs of this brand. Then give its lower limit and upper limit.

Carry your intermediate computations to at least three decimal places. Round your answers to one decimal place.

To find a 95% confidence interval for the true mean lifetime of the light bulbs, we will use the same method as before, employing the formula for the confidence interval of the mean when the population standard deviation is known:

\[
\text{CI} = \bar{x} \pm Z_{\alpha/2} \times \frac{\sigma}{\sqrt{n}}
\]

### Given:
- Sample mean (\(\bar{x}\)) = 486 hours
- Population standard deviation (\(\sigma\)) = 54 hours
- Sample size (\(n\)) = 50
- Confidence level = 95%

### Steps to Solve:

1. **Find the critical value \(Z_{\alpha/2}\) for a 95% confidence level.**
   - For a 95% confidence level, \(\alpha = 1 - 0.95 = 0.05\).
   - \(Z_{\alpha/2}\) corresponds to the value that leaves 2.5% in each tail of the normal distribution, which is approximately 1.96.

2. **Calculate the standard error (SE):**

\[
\text{SE} = \frac{\sigma}{\sqrt{n}} = \frac{54}{\sqrt{50}} \approx 7.636
\]

3. **Calculate the margin of error (ME):**

\[
\text{ME} = Z_{\alpha/2} \times \text{SE} = 1.96 \times 7.636 \approx 14.965
\]

4. **Calculate the confidence interval:**

\[
\text{CI} = 486 \pm 14.965
\]

- **Lower limit**: \(486 - 14.965 \approx 471.0\)
- **Upper limit**: \(486 + 14.965 \approx 501.0\)

### Final Confidence Interval:
The 95% confidence interval for the true mean lifetime of all light bulbs of this brand is \( \textbf{[471.0, 501.0]} \).

Do you want details or have any questions?

---

Here are 8 related questions you might find useful:

1. What is the impact of using a 99% confidence level instead of 95%?
2. How would the confidence interval change with a larger sample size?
3. What assumptions are made when calculating this confidence interval?
4. How does the standard deviation of the population affect the confidence interval?
5. Can you calculate the confidence interval if the sample standard deviation is used instead of the population standard deviation?
6. Why do we use the normal distribution for the critical value?
7. What does it mean if the confidence interval does not contain the hypothesized population mean?
8. How would you interpret the confidence interval in the context of quality control?

**Tip:** A narrower confidence interval provides a more precise estimate but requires a larger sample size or a lower confidence level.

Learn how to calculate a 95% confidence interval for the mean lifetime of a brand of electric light bulbs, based on a sample mean and population standard deviation. Includes step-by-step computations and interpretation. Suitable for college-level statistics.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation