The average hourly wage of workers at a fast food restaurant is $6.17/hr with a variance of  $0.45/hr. Assume that the distribution is normally distributed. If a worker at this fast food restaurant is selected at random, what is the probability that the worker earns more than $6.45/hr? (4 decimal accuracy)


We are given the following information:

- Mean wage $ \mu = 6.17 \, \text{hr} $
- Variance $ \sigma^2 = 0.45 \, \text{hr}^2 $, so standard deviation $ \sigma = \sqrt{0.45} $
- We want to find the probability that a worker earns more than $ 6.45 \, \text{hr} $.

This is a normal distribution problem, and we want to calculate $ P(X > 6.45) $, where $ X $ is the hourly wage.

### Step 1: Standardize the variable to the standard normal distribution (Z-distribution).
We use the formula for the Z-score:
\[
Z = \frac{X - \mu}{\sigma}
\]
Substitute $ X = 6.45 $, $ \mu = 6.17 $, and $ \sigma = \sqrt{0.45} $:

\[
Z = \frac{6.45 - 6.17}{\sqrt{0.45}}
\]

Let's calculate this Z-score first.The Z-score is approximately $ Z = 0.4174 $.

### Step 2: Find the probability corresponding to this Z-score.
We now find the probability that a standard normal variable is greater than this Z-score, i.e., $ P(Z > 0.4174) $. We can look up this Z-score in a standard normal distribution table or use a calculator.

The area to the left of $ Z = 0.4174 $ represents $ P(Z \leq 0.4174) $. To find $ P(Z > 0.4174) $, we subtract this value from 1.

Let's compute the probability.It seems like I can’t do more advanced data analysis right now. Please try again later.

Would you like me to guide you on how to look up the Z-score probability manually, or do you have any other questions?

Here are 5 related questions for further exploration:

1. How can we interpret the Z-score in terms of standard deviations?
2. What is the cumulative distribution function (CDF) for a normal distribution?
3. How would you calculate probabilities for a range of wages, like between $6.00/hr and $6.45/hr?
4. How does variance affect the spread of data in a normal distribution?
5. How can Z-scores be used in hypothesis testing?

**Tip**: When using Z-scores, always subtract from 1 for probabilities greater than the given Z value!

The average hourly wage of workers at a fast food restaurant is $6.17/hr with a variance of $0.45/hr. Assume that the distribution is normally distributed. If a worker at this fast food restaurant is selected at random, what is the probability that the worker earns more than $6.45/hr? (4 decimal accuracy)

Calculate the probability that a fast food worker earns more than $6.45/hr, assuming a normal distribution with a mean wage of $6.17/hr and a variance of $0.45/hr. Learn how to apply Z-scores and probability concepts to solve this real-world problem. Suitable for students in Grades 10-12.

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