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We are given the following information:

- Average hourly wage ($\mu$): $17.42
- Standard deviation ($\sigma$): $2.25
- The wage threshold to calculate the probability for: $20.98

We need to find the probability that a randomly chosen manufacturing worker earns more than $20.98 per hour, assuming a normal distribution.

### Step 1: Calculate the z-score
The formula for the z-score is:

\[
z = \frac{X - \mu}{\sigma}
\]

Where:
- $ X $ is the value of interest ($20.98),
- $ \mu $ is the mean ($17.42),
- $ \sigma $ is the standard deviation ($2.25).

Substituting the values:

\[
z = \frac{20.98 - 17.42}{2.25} = \frac{3.56}{2.25} \approx 1.582
\]

So, the z-score is approximately **1.582**.

### Step 2: Find the probability
The next step is to find the probability of a z-score being greater than 1.582 using standard normal distribution tables or a calculator. 

Looking up the z-score of 1.582 in the standard normal distribution table, we find that the area to the left of this z-score is approximately **0.9439**. 

Thus, the probability to the right (which corresponds to wages greater than $20.98) is:

\[
P(X > 20.98) = 1 - 0.9439 = 0.0561
\]

Therefore, the probability that a randomly chosen worker makes more than $20.98 per hour is approximately **0.0561** or **5.61%**.

### Answers:
1. The z-score is **1.582**.
2. The probability is **0.0561** (or **5.61%**).

---

Do you want further details or have any questions?

Here are 5 additional related questions:
1. How do we interpret the z-score value?
2. What is the cumulative probability for this z-score?
3. How would the probability change if the standard deviation were higher?
4. How can we use the z-score formula to find the wage corresponding to a certain probability?
5. What is the probability that a worker earns less than $20.98?

**Tip:** Always remember that the area under a normal curve represents probabilities, and you can split it into left and right of the z-score!

The average hourly wage for manufacturing workers in the United States is $17.42 per hour, with a standard deviation of $2.25. Assume the data is normally distributed. What is the probability that a randomly chosen manufacturing worker makes more than $20.98 an hour?

Learn how to calculate the probability that a manufacturing worker in the US earns more than $20.98 per hour, based on a normal distribution with a mean of $17.42 and a standard deviation of $2.25. This problem demonstrates how to use z-scores and the empirical rule in statistics.

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