For a standardized psychology examination intended for psychology majors, the historical data show that scores have a mean of and a standard deviation of 170.

Question

For a standardized psychology examination intended for psychology majors, the historical data show that scores have a mean of and a standard deviation of 170. The grading process of this year's exam has just begun. The mean score of the 40 exams graded so far is 540. What is the probability that a sample of 40 exams will have a mean score of 540 or more if the exam scores follow the same distribution as in the past?

Answer 1

use the Central Limit Theorem, which allows us to approximate the distribution of the sample mean.

Given: Population mean (μ) = unknown (given as missing, but it's likely 170 based on the historical data) Population standard deviation (σ) = 170 Sample size (n) = 40 Sample mean (X̄) = 540

Since we don't have the population mean (μ) for this year's exam, we can use the historical data's population mean (170) since it's stated that the exam scores follow the same distribution as in the past.

First, we need to find the standard error of the mean (SE), which is the standard deviation of the sampling distribution of the mean:

SE = σ / √n SE = 170 / √40 ≈ 26.9258 (rounded to 4 decimal places)

Next, we calculate the Z-score for the sample mean of 540:

Z = (X̄ - μ) / SE Z = (540 - 170) / 26.9258 ≈ 14.6935 (rounded to 4 decimal places)

Now, we need to find the probability that the Z-score is greater than 14.6935 using a standard normal distribution table or calculator:

P(Z > 14.6935) ≈ 0 (rounded to 4 decimal places)

Therefore, the probability that a sample of 40 exams will have a mean score of 540 or more is approximately 0%. This suggests that getting a mean score of 540 or more is highly unlikely based on the historical data and the given information.