The mean starting salary offered to graduating students with a certain major in a recent year was $64,245 , with a standard deviation of $3712. A random sample of 85 of that year's graduating students with the major has been selected.

What is the probability that the mean starting salary offered to these 85 students was $64,250 or less?

Carry your intermediate computations to at least four decimal places. Round your answer to at least three decimal places.

To find the probability that the mean starting salary offered to 85 graduating students is $64,250 or less, we can use the Central Limit Theorem, which tells us that the sampling distribution of the sample mean will be approximately normal.

### Given:
- Population mean ($\mu$): $64,245
- Population standard deviation ($\sigma$): $3,712
- Sample size ($n$): 85
- Sample mean ($\bar{x}$): $64,250

### Step 1: Calculate the Standard Error of the Mean

The standard error (SE) of the mean is calculated using the formula:

\[
SE = \frac{\sigma}{\sqrt{n}}
\]

Substitute the given values:

\[
SE = \frac{3712}{\sqrt{85}} = \frac{3712}{9.2195} \approx 402.57
\]

### Step 2: Calculate the Z-score

The Z-score is calculated using the formula:

\[
Z = \frac{\bar{x} - \mu}{SE}
\]

Substitute the given values into the formula:

\[
Z = \frac{64,250 - 64,245}{402.57} = \frac{5}{402.57} \approx 0.0124
\]

### Step 3: Find the Probability

To find the probability that the sample mean is $64,250 or less, we look for $P(Z \leq 0.0124)$.

Using a standard normal distribution table or a calculator, we find the probability for the Z-score:

- The probability $P(Z < 0.0124)$ is approximately 0.5049.

Therefore, the probability that the mean starting salary for these 85 students is $64,250 or less is approximately **0.505**.

Would you like more details or have any questions about this process? Here are some related questions you might find interesting:

1. What is the role of the standard error in determining the probability?
2. How does increasing the sample size affect the standard error?
3. What would be the probability if the sample mean were higher, say $64,300?
4. How does a change in population standard deviation affect the outcome?
5. Why is it necessary to use the Z-score for probability calculations?
6. How accurate is the approximation of the Central Limit Theorem?
7. How can we use the normal distribution to make predictions in real-world scenarios?
8. What tools can assist in calculating probabilities for different distributions?

**Tip:** Familiarize yourself with Z-score tables or use statistical software to quickly find probabilities related to normal distributions. They are crucial for many statistical analyses and hypothesis testing.

Learn how to calculate the probability that the mean starting salary offered to a sample of 85 students is $64,250 or less using the Central Limit Theorem and Z-score calculations. Understand the role of standard error in probability calculations. Suitable for undergraduate students studying statistics and probability.

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