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Sagot :
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.
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.
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