Sure, let's solve the problem step by step using the [tex]\(68\%-95\%-99.7\%\)[/tex] rule and other statistical principles.
### Step 1: Understand the Problem
Given:
- Mean salary ([tex]$\mu$[/tex]): \[tex]$425
- Standard deviation ($[/tex]\sigma[tex]$): \$[/tex]50
- We need to find the probability that a randomly selected worker has a salary between \[tex]$350 and \$[/tex]500.
### Step 2: Calculate the Z-scores
To find the probability that the salary is between \[tex]$350 and \$[/tex]500, we start by converting these salaries to Z-scores using the formula:
[tex]\[ Z = \frac{X - \mu}{\sigma} \][/tex]
where [tex]\( X \)[/tex] is the value of interest, [tex]\( \mu \)[/tex] is the mean, and [tex]\( \sigma \)[/tex] is the standard deviation.
For the lower bound (\[tex]$350):
\[ Z_{\text{lower}} = \frac{350 - 425}{50} = \frac{-75}{50} = -1.5 \]
For the upper bound (\$[/tex]500):
[tex]\[ Z_{\text{upper}} = \frac{500 - 425}{50} = \frac{75}{50} = 1.5 \][/tex]
### Step 3: Use the Standard Normal Distribution
Now, we need to find the probability corresponding to these Z-scores in the standard normal distribution.
The cumulative distribution function (CDF) tells us the probability that a value will fall to the left of a Z-score.
- The probability of Z being less than -1.5 (CDF of -1.5)
- The probability of Z being less than 1.5 (CDF of 1.5)
### Step 4: Find the Probabilities
Using standard normal distribution tables or a calculator with CDF functions:
- The CDF of [tex]\(Z = -1.5\)[/tex] is approximately [tex]\(0.0668\)[/tex]
- The CDF of [tex]\(Z = 1.5\)[/tex] is approximately [tex]\(0.9332\)[/tex]
### Step 5: Calculate the Probability Between Two Z-scores
To find the probability that the salary is between \[tex]$350 and \$[/tex]500, we subtract the CDF of the lower Z-score from the upper Z-score:
[tex]\[ P(350 < X < 500) = \text{CDF}(1.5) - \text{CDF}(-1.5) \][/tex]
So,
[tex]\[ P(350 < X < 500) = 0.9332 - 0.0668 = 0.8664 \][/tex]
Thus, the probability that a worker selected at random makes between \[tex]$350 and \$[/tex]500 is approximately [tex]\(0.8664\)[/tex] or [tex]\(86.64\%\)[/tex].
