To find the probability that a randomly selected worker makes between [tex]$350 and $[/tex]500, we can model this problem using the properties of the normal distribution, given that weekly wages are normally distributed with a mean of [tex]$400 and a standard deviation of $[/tex]50.
Here's the detailed step-by-step solution:
1. Define the parameters of the normal distribution:
- Mean ([tex]$\mu$[/tex]): [tex]$400
- Standard Deviation ($[/tex]\sigma[tex]$): $[/tex]50
2. Convert the actual wages to standard normal (Z) scores:
- The Z-score is calculated by 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.
3. Calculate the Z-score for the lower wage bound ([tex]$350$[/tex]):
- [tex]\[
Z_{\text{lower}} = \frac{350 - 400}{50} = \frac{-50}{50} = -1.0
\][/tex]
4. Calculate the Z-score for the upper wage bound ([tex]$500$[/tex]):
- [tex]\[
Z_{\text{upper}} = \frac{500 - 400}{50} = \frac{100}{50} = 2.0
\][/tex]
5. Determine the probabilities corresponding to these Z-scores:
- The cumulative distribution function (CDF) of a standard normal distribution gives us the probability that a value is less than or equal to a given Z-score.
- Let's denote:
- [tex]\(P(Z \leq -1.0)\)[/tex] as the probability that a value is less than or equal to [tex]\(-1.0\)[/tex]
- [tex]\(P(Z \leq 2.0)\)[/tex] as the probability that a value is less than or equal to [tex]\(2.0\)[/tex]
6. Find the area under the normal curve between these Z-scores:
The probability that a value falls between the Z-scores is given by:
[tex]\[
P(-1.0 \leq Z \leq 2.0) = P(Z \leq 2.0) - P(Z \leq -1.0)
\][/tex]
7. Convert the Z-scores back to the cumulative probability values:
- Probability for [tex]\( P(Z \leq 2.0) \)[/tex] = 0.9772
- Probability for [tex]\( P(Z \leq -1.0) \)[/tex] = 0.1587
8. Calculate the final probability:
[tex]\[
P(350 \leq w \leq 500) = 0.9772 - 0.1587 = 0.8185
\][/tex]
So, the probability that a worker selected at random makes between [tex]$350$[/tex] and [tex]$500$[/tex] is approximately [tex]\(0.8185\)[/tex], or 81.85%.
