The lifetime of a certain brand of electric lightbulb is known to have a standard deviation of 47 hours suppose that a random sample of 90 bulbs of this brand has a mean lifetime of 477 hours. Find a 90% confidence interval for the true mean lifetime of all lightbulbs of this brand then keep its lower limit and upper limit carry your intermediate computations to at least three decimal places around your answers to one decimal place.

To find the 90% confidence interval for the true mean lifetime of all lightbulbs of this brand, we can use the following formula:

\[
\text{Confidence Interval} = \bar{x} \pm z_{\alpha/2} \cdot \left(\frac{\sigma}{\sqrt{n}}\right)
\]

Where:
- \(\bar{x}\) is the sample mean.
- \(z_{\alpha/2}\) is the z-score corresponding to the desired confidence level (90% in this case).
- \(\sigma\) is the population standard deviation.
- \(n\) is the sample size.

### Step 1: Identify the given values
- Sample mean (\(\bar{x}\)) = 477 hours
- Population standard deviation (\(\sigma\)) = 47 hours
- Sample size (\(n\)) = 90

### Step 2: Find the z-score corresponding to a 90% confidence level
For a 90% confidence level, the z-score (\(z_{\alpha/2}\)) is approximately 1.645.

### Step 3: Calculate the standard error of the mean
The standard error (SE) is given by:

\[
\text{SE} = \frac{\sigma}{\sqrt{n}} = \frac{47}{\sqrt{90}}
\]

### Step 4: Calculate the margin of error
The margin of error (ME) is given by:

\[
\text{ME} = z_{\alpha/2} \cdot \text{SE} = 1.645 \cdot \frac{47}{\sqrt{90}}
\]

### Step 5: Calculate the confidence interval
The confidence interval is:

\[
\text{Confidence Interval} = \bar{x} \pm \text{ME}
\]

Let's compute these values.The 90% confidence interval for the true mean lifetime of all lightbulbs of this brand is approximately:

- **Lower limit:** 468.9 hours
- **Upper limit:** 485.1 hours

Would you like more details or have any questions?

Here are some related questions you might find useful:

1. What is the interpretation of a confidence interval in statistics?
2. How does increasing the sample size affect the width of a confidence interval?
3. What would happen to the confidence interval if we used a 95% confidence level instead?
4. How is the z-score determined for different confidence levels?
5. What assumptions are necessary for constructing a confidence interval for a mean?
6. What is the difference between a t-distribution and a z-distribution?
7. How can you reduce the margin of error in a confidence interval?
8. Why is it important to carry intermediate computations to at least three decimal places?

**Tip:** When constructing confidence intervals, always ensure the sample size is sufficiently large to provide an accurate estimate of the population parameter.

Learn how to calculate a 90% confidence interval for the mean lifetime of a brand of lightbulbs. Includes step-by-step computations using sample mean, standard deviation, and sample size. Ideal for undergraduate-level statistics students.

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