Final answer:
To determine the probability that market returns are between -10% and 30%, we use the Empirical Rule and z-scores. The returns fall within one standard deviation of the mean, which corresponds to approximately 68% of values in a normal distribution.
Step-by-step explanation:
The student's question involves finding the probability that the market return will be between -10% and 30% given that the market returns are normally distributed with a 10% mean and a 20% standard deviation. To solve this, we can use the Empirical Rule and understanding of z-scores in a normal distribution.
According to the Empirical Rule, approximately 68% of values lie within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations. For the given mean of 10%, a return of -10% is one standard deviation below the mean (since the standard deviation is 20%), and a return of 30% is one standard deviation above the mean. Hence, approximately 68% of the values lie between -10% and 30%.
The z-scores associated with -10% and 30% are -1 and +1, respectively. Since we know that roughly 68% of values in a normal distribution fall between z-scores of -1 and 1, we can conclude that the probability of the market return being between -10% and 30% is approximately 68%.