Final answer:
To calculate the probability of a sample mean IQ between 85 and 125, find the standard error, convert the IQ scores to Z-scores, and subtract the probability of the mean being less than 85 from the probability of being less than 125.
Step-by-step explanation:
The task is to find the probability that a randomly selected sample of 20 adults has a mean IQ score between 85 and 125, given that IQ scores are normally distributed with a mean of 100 and a standard deviation of 15.
To calculate this, we can use the Central Limit Theorem which states that the distribution of sample means will be approximately normally distributed if the sample size is large enough (generally n > 30 is sufficient, but n=20 can be considered adequate for our purposes).
First, we calculate the standard error of the mean which is the standard deviation divided by the square root of the sample size (n). Standard Error (SE) = 15 / sqrt(20). Then, we standardize the range of IQ scores to Z-scores, where the Z-score formula is (X - mean) / SE.
After calculating Z-scores for 85 and 125, we use standard normal distribution tables or a calculator to find the probability corresponding to those Z-scores.
To find the probability that the mean score is between 85 and 125, we compute the probabilities of the mean being less than 125 (P(X < 125)) and less than 85 (P(X < 85)), then subtract the latter from the former. The result gives us the probability that the sample mean falls within the desired range.