For certain workers, the mean wage is $6.50/hr, with a standard deviation of $0.75. If a worker is chosen at random, what is the probability that the worker's wage is between $6.25 and $8.25? Assume a normal distribution of wages.



Answer :

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:

[tex]\begin{gathered} \Rightarrow z=\frac{wage-\operatorname{mean}\text{ wage}}{\text{standard deviation wage}} \\ O\text{therwise written as:} \\ z=\frac{x-\mu}{\sigma} \end{gathered}[/tex]

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:

[tex]\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=\frac{4.25-6.5}{0.75}=-\frac{2.25}{0.75}=-3 \\ \text{Thus, when wage is 8.75 dollars, we have:} \\ \Rightarrow z=\frac{8.75-6.5}{0.75}=\frac{2.25}{0.75}=3 \end{gathered}[/tex]

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

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