Step 1 :Given that the sample mean (x_bar) is 498, the standard deviation (sigma) is 47, the sample size (n) is 90, and the Z-score for a 99% confidence interval (Z) is 2.576.
Step 2 :First, calculate the margin of error using the formula: \(Z \times \frac{\sigma}{\sqrt{n}}\).
Step 3 :Substitute the given values into the formula to get the margin of error: \(2.576 \times \frac{47}{\sqrt{90}} = 12.762109362396867\).
Step 4 :Next, calculate the lower limit of the confidence interval using the formula: \(x_{bar} - \text{margin of error}\).
Step 5 :Substitute the values into the formula to get the lower limit: \(498 - 12.762109362396867 = 485.23789063760313\).
Step 6 :Then, calculate the upper limit of the confidence interval using the formula: \(x_{bar} + \text{margin of error}\).
Step 7 :Substitute the values into the formula to get the upper limit: \(498 + 12.762109362396867 = 510.76210936239687\).
Step 8 :Round the lower and upper limits to one decimal place to get the final answer: lower limit is 485.2 and upper limit is 510.8.
Step 9 :The 99% confidence interval for the true mean lifetime of all light bulbs of this brand is approximately \(\boxed{[485.2, 510.8]}\) hours.