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Morgan sampled 101 students and calculated an average of 6.5 hours of sleep each night with a standard deviation of 2.14. Using a [tex][tex]$90 \%$[/tex][/tex] confidence level, she also found that [tex][tex]$t^\ \textless \ em\ \textgreater \ =1.660$[/tex][/tex].

The confidence interval is given by:
[tex]\bar{x} \pm t^\ \textless \ /em\ \textgreater \ \frac{s}{\sqrt{n}}[/tex]

A [tex][tex]$90 \%$[/tex][/tex] confidence interval calculates that the average number of hours of sleep for working college students is between __________ hours. Answer choices are rounded to the hundredths place.

a.) 6.46 and 6.85
b.) 6.15 and 6.85
c.) 6.46 and 6.54


Sagot :

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.