To find the 90% confidence interval for the average number of hours of sleep for working college students, we need to go through the following steps:
1. Identify the given values:
- Sample size ([tex]\( n \)[/tex]): 101
- Sample mean ([tex]\( \bar{x} \)[/tex]): 6.5 hours
- Sample standard deviation ([tex]\( s \)[/tex]): 2.14
- [tex]\( t^* \)[/tex]: 1.660 (for a 90% confidence level)
2. Calculate the standard error:
The standard error (SE) of the mean is calculated using the formula:
[tex]\[
SE = \frac{s}{\sqrt{n}}
\][/tex]
Plugging in the given values:
[tex]\[
SE = \frac{2.14}{\sqrt{101}} \approx 0.2129379587049377
\][/tex]
3. Calculate the margin of error:
The margin of error (ME) is calculated using the formula:
[tex]\[
ME = t^* \times SE
\][/tex]
Plugging in the values:
[tex]\[
ME = 1.660 \times 0.2129379587049377 \approx 0.35347701145019655
\][/tex]
4. Calculate the confidence interval:
The 90% confidence interval is given by:
[tex]\[
\bar{x} \pm ME
\][/tex]
- Lower bound:
[tex]\[
\bar{x} - ME = 6.5 - 0.35347701145019655 \approx 6.146522988549804
\][/tex]
- Upper bound:
[tex]\[
\bar{x} + ME = 6.5 + 0.35347701145019655 \approx 6.853477011450196
\][/tex]
Therefore, the 90% confidence interval for the average number of hours of sleep for working college students is between approximately [tex]\( 6.15 \)[/tex] hours and [tex]\( 6.85 \)[/tex] hours.
The correct answer, rounded to the hundredths place, is (b) 6.15 and 6.85.
