To obtain the probability that the worker's wage is between $4.25 and $8.75 (assuming a normal distribution of wages) the following steps are necessary:
Step 1: Convert the wages given to a z-score using the following formula:
![\begin{gathered} \Rightarrow z=\frac{wage-\operatorname{mean}\text{ wage}}{\text{standard deviation wage}} \\ O\text{therwise written as:} \\ z=(x-\mu)/(\sigma) \end{gathered}]()
Step 2: Convert the $4.25 and $8.75 to their respective z-score using the mean of $6.50 and standard deviation of $0.75, as shown below:
![\begin{gathered} z=\frac{wage-\operatorname{mean}\text{ wage}}{\text{standard deviation wage}} \\ \text{Thus, when wage is 4.25 dollars, we have:} \\ \Rightarrow z=(4.25-6.5)/(0.75)=-(2.25)/(0.75)=-3 \\ \text{Thus, when wage is 8.75 dollars, we have:} \\ \Rightarrow z=(8.75-6.5)/(0.75)=(2.25)/(0.75)=3 \end{gathered}]()
Thus, we are to find the probability that the wages of the workers lie between the z-score of -3 and +3.
To do this, we have to refer to the normal distribution curve, shown below:
Step 3: We now interpret the graph as follows:
According to the graph, we can see that the 99.73% of the area of the curve lies between the z-scores of -3 and +3.
This means that the probability that the wages of the workers lie between the z-score of -3 and +3 (or that the worker's wage is between $4.25 and $8.75) is 99.73% or 0.9973