Construct a probability distribution for the number of overtime hours PPG operators worked in one week, per employee. Round [tex][tex]$P(x)$[/tex][/tex] to three decimal places.

\begin{tabular}{|c|c|c|}
\hline Overtime hours, [tex]$x$[/tex] & Employees, [tex]$f$[/tex] & [tex]$P(x)$[/tex] \\
\hline 0 & 7 & \\
\hline 1 & 14 & \\
\hline 2 & 28 & \\
\hline 3 & 52 & \\
\hline 4 & 35 & \\
\hline 5 & & \\
\hline
\end{tabular}



Answer :

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.