This problem involves using the standard normal distribution to calculate probabilities. Let’s break it down step-by-step:

Given:

• Mean ($\mu$) = $50,824
• Standard deviation ($\sigma$) = $4,294
• We need to find $P(X > 55,000)$.

Solution:

Step 1: Calculate the z-score.

The z-score formula is: $z = \frac{X - \mu}{\sigma}$

Substitute the values: $z = \frac{55,000 - 50,824}{4,294} = \frac{4,176}{4,294} \approx 0.97$

Step 2: Find the probability.

Using the standard normal distribution table or a calculator, find the cumulative probability corresponding to $z = 0.97$.

From the table: $P(Z \leq 0.97) \approx 0.8340$

Since we are looking for $P(X > 55,000)$, we calculate: $P(X > 55,000) = 1 - P(Z \leq 0.97) = 1 - 0.8340 = 0.1660$

Final Answer:

$P(X > 55,000) = 0.1660$

Let me know if you'd like a detailed explanation of any step or need further help!

Related Questions:

1. What is the probability that a graduate makes less than $45,000?
2. What is the probability that a graduate earns between $48,000 and $52,000?
3. How do we interpret the standard deviation in this context?
4. What would happen to the z-score if the mean salary increased to $52,000?
5. How do we use the standard normal distribution to solve real-world problems?

Tip:

Always sketch a bell curve for problems involving the normal distribution—it helps visualize the problem and avoid mistakes!