Absolutely, let's go through the steps needed to calculate the margin of error for a 90% confidence level given the provided data:
1. Identify the mean and standard deviation:
- Mean ([tex]\(\mu\)[/tex]): 24 hours
- Standard Deviation ([tex]\(\sigma\)[/tex]): 2.9 hours
2. Determine the z-score for the given confidence level:
For a 90% confidence level, the z-score ([tex]\(z^*\)[/tex]) is 1.645 (as given in the table).
3. Calculate the margin of error:
The formula for the margin of error (ME) is:
[tex]\[
\text{ME} = z^* \cdot \sigma
\][/tex]
4. Substitute the given values:
- [tex]\(z^*\)[/tex]: 1.645
- [tex]\(\sigma\)[/tex]: 2.9
[tex]\[
\text{ME} = 1.645 \times 2.9
\][/tex]
5. Perform the multiplication:
[tex]\[
\text{ME} = 4.7705
\][/tex]
6. Round the margin of error to the nearest tenth:
[tex]\[
\text{ME} \approx 4.8
\][/tex]
Therefore, the margin of error, when rounded to the nearest tenth, is [tex]\(4.8\)[/tex] hours.
