Answer :
To calculate the 99% confidence interval for the mean lifetime of the company's D-batteries, follow these steps:
1. Calculate the Sample Mean:
The sample mean of the given data is the average lifetime of the batteries.
[tex]\[
\text{Sample Mean} = 39.85 \text{ hours (rounded to 2 decimal places)}
\][/tex]
2. Calculate the Sample Standard Deviation:
The sample standard deviation measures the dispersion of the battery lifetimes from the sample mean.
[tex]\[
\text{Sample Standard Deviation} = 3.67 \text{ hours (rounded to 2 decimal places)}
\][/tex]
3. Determine the Sample Size:
The sample size [tex]\( n \)[/tex] is the number of batteries used in the study.
[tex]\[
n = 60
\][/tex]
4. Define the Confidence Level:
The confidence level is 99%.
5. Calculate the Z-value for the 99% Confidence Level:
The Z-value corresponding to a 99% confidence level is typically 2.576.
6. Calculate the Standard Error:
The standard error of the mean is calculated using the sample standard deviation and the square root of the sample size.
[tex]\[
\text{Standard Error} = \frac{\text{Sample Standard Deviation}}{\sqrt{n}} = \frac{3.67}{\sqrt{60}} \approx 0.47
\][/tex]
7. Calculate the Margin of Error:
The margin of error is determined by multiplying the Z-value by the standard error.
[tex]\[
\text{Margin of Error} = Z \times \text{Standard Error} = 2.576 \times 0.47 \approx 1.22 \text{ hours (rounded to 2 decimal places)}
\][/tex]
8. Calculate the Confidence Interval:
Finally, the confidence interval is calculated by subtracting and adding the margin of error from/to the sample mean.
[tex]\[
\text{Lower Confidence Limit} = \text{Sample Mean} - \text{Margin of Error} = 39.85 - 1.22 = 38.63 \text{ hours (rounded to 2 decimal places)}
\][/tex]
[tex]\[
\text{Upper Confidence Limit} = \text{Sample Mean} + \text{Margin of Error} = 39.85 + 1.22 = 41.07 \text{ hours (rounded to 2 decimal places)}
\][/tex]
Therefore, the 99% confidence interval for the mean lifetime of the company's D-batteries is:
Lower Confidence Limit = 38.63 hours
Upper Confidence Limit = 41.07 hours
1. Calculate the Sample Mean:
The sample mean of the given data is the average lifetime of the batteries.
[tex]\[
\text{Sample Mean} = 39.85 \text{ hours (rounded to 2 decimal places)}
\][/tex]
2. Calculate the Sample Standard Deviation:
The sample standard deviation measures the dispersion of the battery lifetimes from the sample mean.
[tex]\[
\text{Sample Standard Deviation} = 3.67 \text{ hours (rounded to 2 decimal places)}
\][/tex]
3. Determine the Sample Size:
The sample size [tex]\( n \)[/tex] is the number of batteries used in the study.
[tex]\[
n = 60
\][/tex]
4. Define the Confidence Level:
The confidence level is 99%.
5. Calculate the Z-value for the 99% Confidence Level:
The Z-value corresponding to a 99% confidence level is typically 2.576.
6. Calculate the Standard Error:
The standard error of the mean is calculated using the sample standard deviation and the square root of the sample size.
[tex]\[
\text{Standard Error} = \frac{\text{Sample Standard Deviation}}{\sqrt{n}} = \frac{3.67}{\sqrt{60}} \approx 0.47
\][/tex]
7. Calculate the Margin of Error:
The margin of error is determined by multiplying the Z-value by the standard error.
[tex]\[
\text{Margin of Error} = Z \times \text{Standard Error} = 2.576 \times 0.47 \approx 1.22 \text{ hours (rounded to 2 decimal places)}
\][/tex]
8. Calculate the Confidence Interval:
Finally, the confidence interval is calculated by subtracting and adding the margin of error from/to the sample mean.
[tex]\[
\text{Lower Confidence Limit} = \text{Sample Mean} - \text{Margin of Error} = 39.85 - 1.22 = 38.63 \text{ hours (rounded to 2 decimal places)}
\][/tex]
[tex]\[
\text{Upper Confidence Limit} = \text{Sample Mean} + \text{Margin of Error} = 39.85 + 1.22 = 41.07 \text{ hours (rounded to 2 decimal places)}
\][/tex]
Therefore, the 99% confidence interval for the mean lifetime of the company's D-batteries is:
Lower Confidence Limit = 38.63 hours
Upper Confidence Limit = 41.07 hours