Question

Weekly wages at a certain factory arenormally distributed with a mean of$400 and a standard deviation of $50.Find the probability that a workerselected at random makes between$500 and $550.

Answer 1

The Solution:

Step 1:

We shall state the formula for calculating Z-score.

`\begin{gathered} Z=(X-\mu)/(\sigma) \\ \text{Where X}=5\text{00 ( for lower limit) and X=550 for upper limit.} \\ \mu=400 \\ \sigma=50 \end{gathered}`

Step 2:

We shall substitute the above values in the formula.

`\begin{gathered} (500-400)/(50)\leq P(Z)\leq(550-400)/(50) \\ \\ (100)/(50)\leq P(Z)\leq(150)/(50) \\ \\ 2\leq P(Z)\leq3 \end{gathered}`

Step 3:

We shall read the respective probabilities from the Z score distribution tables.

From the Z-score tables,

P(3) = 99.9 %

P(2) = 97.7 %

Step 4:

The Conclusion:

The probability that a worker selected makes between $500 and $550 is obtained as below:

`\text{Prob}(500\leq Z\leq550)=99.9-97.7\text{ = 2.2 \%}`

Therefore, the required probability is 2.2 %