To find a 99% confidence interval for the population mean of light bulbs, we'll use the formula for the confidence interval, which is given by:
\[ \text{CI} = \bar{x} \pm z \left( \frac{\sigma}{\sqrt{n}} \right) \]
Where:
- \(\bar{x}\) is the sample mean,
- \(z\) is the z-score corresponding to the desired confidence level,
- \(\sigma\) is the population standard deviation,
- \(n\) is the sample size.
Given:
- \(\bar{x} = 780\) hours (sample mean),
- \(\sigma = 40\) hours (standard deviation),
- \(n = 36\) (sample size),
- We want a 99% confidence interval.
Step 1: Find the z-score for 99% confidence
For a 99% confidence level, we need the z-score that corresponds to the critical value. The critical z-score can be found using a standard normal distribution table, or using statistical software or an online calculator.
For a 99% confidence level, the z-score (two-tailed) is approximately:
\[ z \approx 2.576 \]
Step 2: Calculate the standard error (SE)
The standard error (SE) is calculated using the formula:
\[ SE = \frac{\sigma}{\sqrt{n}} = \frac{40}{\sqrt{36}} = \frac{40}{6} \approx 6.67 \]
Step 3: Calculate the margin of error (ME)
Now, we compute the margin of error using the z-score and the standard error:
\[ ME = z \cdot SE = 2.576 \cdot 6.67 \approx 17.24 \]
Step 4: Calculate the confidence interval
Now we can calculate the confidence interval:
\[ \text{Lower limit} = \bar{x} - ME = 780 - 17.24 \approx 762.76 \] \[ \text{Upper limit} = \bar{x} + ME = 780 + 17.24 \approx 797.24 \]
Conclusion
The 99% confidence interval for the population mean life of all bulbs supplied by this company is approximately:
\[ \boxed{(762.76, 797.24)} \text{ hours} \]
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