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.