To determine whether the mean number of hours worked per week by men in the sample differs from the 40-hour standard using a single-sample [tex]\( t \)[/tex]-statistic, follow these steps:
### Step 1: Gather the given information
- Sample size ([tex]\( n \)[/tex]): 60
- Sample mean ([tex]\( \bar{x} \)[/tex]): 42.31 hours
- Population mean (standard mean) ([tex]\( \mu \)[/tex]): 40 hours
- Sample standard deviation ([tex]\( s \)[/tex]): 10.00 hours
### Step 2: Calculate the standard error of the mean ([tex]\( SE \)[/tex])
The standard error of the mean is given by:
[tex]\[
SE = \frac{s}{\sqrt{n}}
\][/tex]
Where:
- [tex]\( s \)[/tex] is the sample standard deviation
- [tex]\( n \)[/tex] is the sample size
Substituting the given values:
[tex]\[
SE = \frac{10.00}{\sqrt{60}}
\][/tex]
[tex]\[
SE \approx 1.2909944487358056
\][/tex]
### Step 3: Calculate the [tex]\( t \)[/tex]-statistic
The [tex]\( t \)[/tex]-statistic is calculated using the formula:
[tex]\[
t = \frac{\bar{x} - \mu}{SE}
\][/tex]
Where:
- [tex]\( \bar{x} \)[/tex] is the sample mean
- [tex]\( \mu \)[/tex] is the population mean
- [tex]\( SE \)[/tex] is the standard error
Substituting the given values:
[tex]\[
t = \frac{42.31 - 40}{1.2909944487358056}
\][/tex]
[tex]\[
t \approx 1.7893183059478284
\][/tex]
### Conclusion
The value of the single-sample [tex]\( t \)[/tex]-statistic is approximately [tex]\( 1.79 \)[/tex].
