Final answer:
Using the Central Limit Theorem, we find there's a 97.64% probability that the average stipend for 25 full-time PhD students will be between $15,000 and $16,500.
Step-by-step explanation:
To solve the probability problem, we need to use the Central Limit Theorem since we're dealing with the average stipend of 25 students. We assume that the distribution of stipends is normal or approximately normal. The stipend average is $15,837 per year with a standard deviation of $1,600.
First, we find the standard error of the mean by dividing the standard deviation by the square root of the number of students:
Standard Error (SE) = $1,600 / √25 = $320.
Next, we convert the requested stipend range to z-scores. The formula for calculating a z-score is Z = (X - μ) / SE, where X is the value in question, μ is the mean, and SE is the standard error.
- For $15,000: Z = ($15,000 - $15,837) / $320 = -2.62
- For $16,500: Z = ($16,500 - $15,837) / $320 = 2.07
We then look up these z-scores on the standard normal distribution table to find the probabilities:
- P(Z < -2.62) is approximately 0.0044
- P(Z < 2.07) is approximately 0.9808
Finally, calculate the probability of the average stipend being between $15,000 and $16,500 for 25 students by subtracting the smaller probability from the larger one:
Probability = 0.9808 - 0.0044 = 0.9764.
Thus, there's about a 97.64% chance that the average stipend for 25 full-time PhD students will fall between $15,000 and $16,500.