Final answer:
To determine the probability that the mean salary of the sample is less than $57,500, calculate the z-score using the population mean, standard deviation, and sample size and then find the associated probability. Interpretation depends on the computed probability; a larger percentage means the event is not unusual, whereas a smaller one means it is unusual.
Step-by-step explanation:
To find the probability that the mean salary of the sample is less than $57,500, we need to use the Central Limit Theorem (CLT). Given that the population mean annual salary is $60,000, the standard deviation is $5,900, and the sample size is 42, we can calculate the z-score for a sample mean of $57,500.
First, we find the standard error (SE) of the mean:
SE = σ / √ n = $5,900 / √ 42
Next, we calculate the z-score:
z = (X - μ) / SE = ($57,500 - $60,000) / SE
After calculating the z-score, we look up the probability associated with that z-score in the standard normal distribution table or use technology (like a calculator or statistical software) to find the probability. Let's say P(z) is the probability of the z-score.
Interpreting the results depends on P(z). If P(z) is relatively large (e.g., around 0.30 or 30%), it indicates that about 30% of the samples could have a mean salary less than $57,500, and this would not be an unusual event (answer A). If P(z) is very small (e.g., around 0.003 or 0.3%), it indicates that only 0.3% of the samples are expected to have a mean salary less than $57,500, which would be an unusual event (answers B and C). Answer D may be discarded as it repeats the possibility of A without giving us a rationale for why it would be considered unusual.
Without the exact probability from the z-score, we cannot choose between answers A, B, or C, but we can eliminate D based on the reasoning given above.