Final answer:
To find the probability that the average of two correlated normally distributed random variables falls within a certain range, we calculate the mean and variance of their average, determine the z-scores for the range limits, and then find the corresponding probabilities. Option b) 0.6304 is the correct answer.
Step-by-step explanation:
The question involves finding the probability that the average of two correlated normally distributed random variables lies between two specified values. Given that X1 and X2 are normally distributed random variables with a mean (μ) of 6 and a variance (σ²) of 25, and they have a population correlation (ρ) of 0.3, we first need to determine the distribution of their average. The average of two normally distributed variables is also normally distributed, with mean μ and variance σ²/n.
In this case, the average μ of X1 and X2 is 6, as it is the average of their means. The variance of the average will be reduced because we are taking the mean of two variables; it's σ²/n plus the covariance (which is σ² times the correlation ρ for two variables). So the variance of the average is (25/2) + (25 * 0.3). Hence, the standard deviation (σ) of the average is the square root of this variance.
Next, we calculate the z-scores for the boundaries 3 and 7 and use the standard normal distribution to find the probabilities associated with these z-scores. These probabilities will then be used to find the probability that the average lies between 3 and 7 by subtraction.
After calculating the required z-scores and looking up the corresponding probabilities from the standard normal distribution table, we find that the probability lies closest to 0.6304, making option b) 0.6304 the correct answer.