To construct a 99% confidence interval for the population mean, we’ll calculate each part step-by-step.

Given values:

• Sample size, $n = 30$
• Sample mean, $\bar{x} = 780$ hours
• Population standard deviation, $\sigma = 40$ hours

Steps:

1. Standard Error (SE): $SE = \frac{\sigma}{\sqrt{n}} = \frac{40}{\sqrt{30}} \approx 7.302$

2. Point Estimate:

   • The point estimate for the population mean is simply the sample mean, so: $\text{Point Estimate} = 780 \text{ hours}$

3. Critical Value (z*) for 99% Confidence:

   • For a 99% confidence level, the critical value $z\*$ corresponds to 2.576 (from standard z-tables for a two-tailed test at 99% confidence).
   • Thus: $z\* \approx 2.576$

4. Margin of Error (ME): $ME = z\* \times SE = 2.576 \times 7.302 \approx 18.80$

5. Confidence Interval: $\text{Confidence Interval} = \bar{x} \pm ME = 780 \pm 18.80$

   • Lower limit: $780 - 18.80 = 761.2$
   • Upper limit: $780 + 18.80 = 798.8$

Summary of Answers:

• Sample size, $n$ = 30
• Sample mean, $\bar{x}$ (hours) = 780
• Population standard deviation, $\sigma$ (hours) = 40
• Standard error (hours) = 7.302 (correct to 3 decimal places)
• Point estimate (hours) = 780
• Critical value, $z\*$ = 2.576 (correct to 3 decimal places)
• Margin of error (hours) = 18.80 (correct to 2 decimal places)
• Confidence interval: We are 99% confident that the population mean is between 761.2 hours and 798.8 hours (both correct to 1 decimal place).

Would you like further details or have any questions about this calculation?

Further Related Questions:

1. How would the confidence interval change with a 95% confidence level?
2. What impact does increasing the sample size have on the confidence interval width?
3. Why is the z-value used instead of the t-value in this calculation?
4. How does the confidence interval change if the standard deviation increases?
5. What would be the margin of error with a sample size of 50 bulbs?

Tip:

Confidence intervals provide a range within which we expect the true population mean to lie, given a certain level of confidence, offering insight into data reliability and precision.