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Imagine you wish to determine whether the mean number of hours worked per week by men in a sample differs from the [tex]$40$[/tex]-hr standard. Use the information below to calculate the value of the single-sample [tex]$t$[/tex] statistic.

[tex]\[
\begin{tabular}{|l|l|}
\hline
\textbf{Weekly Hours Worked By Men} & \\
\hline
Range & 60 \\
\hline
Minimum value & 5 \\
\hline
Maximum value & 65 \\
\hline
Mean & 42.31 \\
\hline
Variance & 100.00 \\
\hline
Standard deviation & 10.00 \\
\hline
Sum & 1,227 \\
\hline
\end{tabular}
\][/tex]



Answer :

GinnyAnswer
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].

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