Final answer:
To find the probability that the mean gain for a random sample of 35 years was between 500 and 700, we need to standardize the values and use a standard normal distribution table or a calculator. The probability is 0.2936, which means that approximately 29.37% of samples of 35 years will have an annual mean gain between 500 and 700.
Step-by-step explanation:
To find the probability that the mean gain for a random sample of 35 years was between 500 and 700, we need to standardize the values using the formula z = (x - μ) / (σ / sqrt(n)). Here, x = 500, μ = 651, σ = 1540, and n = 35. Plugging in these values, we get z1 = (500 - 651) / (1540 / sqrt(35)) and z2 = (700 - 651) / (1540 / sqrt(35)). Using a standard normal distribution table or a calculator, we can find the probabilities associated with these z-values. The probability that the mean gain for the sample was between 500 and 700 is the difference between these two probabilities.
Using a standard normal distribution table or a calculator, we find that the z1-value corresponds to a probability of 0.1963 and the z2-value corresponds to a probability of 0.4899. Therefore, the probability that the mean gain for the sample was between 500 and 700 is 0.4899 - 0.1963 = 0.2936 (rounded to four decimal places).
The result can be interpreted as follows: Approximately 29.37% of samples of 35 years will have an annual mean gain between 500 and 700.