To construct the 99% confidence interval for the mean battery-life of the new tablet computer, we follow these steps:
1. Identify the Sample Statistics:
- Sample Mean (\(\bar{x}\)) = 6 hours
- Sample Standard Deviation (\(s\)) = 1.5 hours
- Sample Size (\(n\)) = 20
2. Calculate the Standard Error of the Mean:
The standard error (SE) is calculated using the formula:
[tex]\[
SE = \frac{s}{\sqrt{n}}
\][/tex]
Substituting the values:
[tex]\[
SE = \frac{1.5}{\sqrt{20}} \approx 0.3354
\][/tex]
3. Determine the Critical Value:
For a 99% confidence interval and a sample size of 20, we use the t-distribution. The critical t-value corresponding to a 99% confidence level for 19 degrees of freedom (sample size - 1) is approximately 2.861.
4. Calculate the Margin of Error:
The margin of error (ME) is calculated using the formula:
[tex]\[
ME = t_{\frac{\alpha}{2}} \times SE
\][/tex]
Substituting the values:
[tex]\[
ME = 2.861 \times 0.3354 \approx 0.9596
\][/tex]
5. Construct the Confidence Interval:
The confidence interval is given by:
[tex]\[
\bar{x} \pm ME
\][/tex]
Substituting the values:
[tex]\[
6 \pm 0.9596
\][/tex]
Therefore, the 99% confidence interval is approximately:
[tex]\[
(5.0404, 6.9596)
\][/tex]
Matching this calculation with the given options, the correct form for the 99% confidence interval is:
[tex]\[
CI_{.99}(\mu) = 6 \pm 2.861(0.3354)
\][/tex]
So, the correct choice is:
[tex]\[
CI_{.99}(\mu) = 6 \pm 2.861(0.3354)
\][/tex]
