Sure, let's construct a probability distribution for the number of overtime hours PPG operators worked in one week, per employee. Here are the steps to create this distribution:
1. Calculate the Total Number of Employees:
To find the probability distribution [tex]\( P(x) \)[/tex], we first need to calculate the total number of employees. The total number of employees is the sum of employees for each category of overtime hours.
[tex]\[
\text{Total employees} = 7 + 14 + 28 + 52 + 35 = 136
\][/tex]
2. Calculate the Probability [tex]\( P(x) \)[/tex] for Each Overtime Hour:
For each overtime hour [tex]\( x \)[/tex], [tex]\( P(x) \)[/tex] is calculated by dividing the number of employees [tex]\( f \)[/tex] for that overtime hour by the total number of employees.
[tex]\[
P(x) = \frac{f}{\text{Total employees}}
\][/tex]
We need to round each probability to three decimal places.
- For 0 overtime hours:
[tex]\[
P(0) = \frac{7}{136} \approx 0.051
\][/tex]
- For 1 overtime hour:
[tex]\[
P(1) = \frac{14}{136} \approx 0.103
\][/tex]
- For 2 overtime hours:
[tex]\[
P(2) = \frac{28}{136} \approx 0.206
\][/tex]
- For 3 overtime hours:
[tex]\[
P(3) = \frac{52}{136} \approx 0.382
\][/tex]
- For 4 overtime hours:
[tex]\[
P(4) = \frac{35}{136} \approx 0.257
\][/tex]
3. Complete the Table:
Now, let's fill in the table with the calculated probabilities:
[tex]\[
\begin{array}{|c|c|c|}
\hline \text{Overtime hours, } x & \text{Employees, } f & P(x) \\
\hline 0 & 7 & 0.051 \\
\hline 1 & 14 & 0.103 \\
\hline 2 & 28 & 0.206 \\
\hline 3 & 52 & 0.382 \\
\hline 4 & 35 & 0.257 \\
\hline
\end{array}
\][/tex]
And that provides us with our probability distribution for the number of overtime hours worked by PPG operators in one week, rounded to three decimal places.
