To calculate the probability that the mean gain for a sample of 33 years falls between 500 and 800, we can use the Central Limit Theorem. The Central Limit Theorem states that the distribution of sample means tends to be approximately normal, regardless of the shape of the population, when the sample size is large enough.
Given that the population standard deviation (sigma) is 1539 and the sample size (n) is 33, we can use the following formula to calculate the standard error (SE) of the sample mean:
SE = sigma / sqrt(n)
SE = 1539 / sqrt(33)
Next, we can calculate the z-scores corresponding to the lower and upper bounds of the desired range. The z-score formula is given by:
z = (x - μ) / SE
For the lower bound (500), the z-score is:
z_lower = (500 - μ) / SE
For the upper bound (800), the z-score is:
z_upper = (800 - μ) / SE
Now, we need to consult a standard normal distribution table or use statistical software to find the corresponding probabilities for these z-scores.
Let's denote the probability for the lower bound as P1 and the probability for the upper bound as P2. The probability of the mean gain for the sample falling between 500 and 800 is:
P = P2 - P1
Please note that we need to know the mean annual gain of the Dow Jones Industrial Average to calculate the z-scores and find the probabilities accurately.