To determine the margin of error for the mean number of hours of sleep with a 95% confidence level, we go through the following steps:
1. Identify the given information:
- Sample size ([tex]\(n\)[/tex]) = 324 adults
- Mean number of hours of sleep ([tex]\(\bar{x}\)[/tex]) = 7.5 hours
- Standard deviation ([tex]\(\sigma\)[/tex]) = 1.6 hours
- [tex]\(z^*\)[/tex]-score for a 95% confidence level = 1.96 (from the z-score table provided)
2. Calculate the standard error of the mean (SE):
The standard error is given by the formula:
[tex]\[
SE = \frac{\sigma}{\sqrt{n}}
\][/tex]
Plugging in the values:
[tex]\[
SE = \frac{1.6}{\sqrt{324}} = \frac{1.6}{18} \approx 0.0889
\][/tex]
3. Calculate the margin of error (ME):
The margin of error is given by the formula:
[tex]\[
ME = z^* \cdot SE
\][/tex]
Using the [tex]\(z^*\)[/tex]-score for a 95% confidence level:
[tex]\[
ME = 1.96 \cdot 0.0889 \approx 0.1742
\][/tex]
4. Round the margin of error to the nearest hundredth:
[tex]\[
ME \approx 0.17
\][/tex]
So, the margin of error for the mean number of hours of sleep, with a 95% confidence level, is approximately 0.17 hours.
