Week 4 Homework
Question 7 of $\mathbf{2 0}$ (1 point) | Question Attempt: 1 of Unlimited
Ayesha
4
5
6
7
8
9
10
Español
12
13
14
The lifetime of a certain brand of electric light bulb 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 498 hours. Find a $99 \%$ confidence interval for the true mean lifetime of all light bulbs of this brand. Then give its lower limit and upper limit.

Carry your intermediate computations to at least three decimal places. Round your answers to one decimal place. (If necessary, consult a list of formulas.)
Lower limit:
Upper limit:

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.