To solve this, we need to standardize the given value (10.3%) using the Z-score formula:

$Z = \frac{X - \mu}{\sigma}$

Where:

• $X = 10.3\%$
• $\mu = 11.5\%$
• $\sigma = 2.7\%$

Now, let's calculate the Z-score:

$Z = \frac{10.3 - 11.5}{2.7} = \frac{-1.2}{2.7} \approx -0.444$

Next, we find the cumulative probability corresponding to this Z-score using the standard normal distribution.

For $Z = -0.444$, the cumulative probability is approximately 0.328.

Thus, the probability that the mutual fund will have a yearly gain of less than 10.3% is 0.328.

Let me know if you'd like further explanation or more questions!

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Further questions:

1. How would the probability change if the mean gain were 12% instead of 11.5%?
2. What is the Z-score for a yearly gain of 8%?
3. How do you interpret the Z-score in the context of this problem?
4. What would happen to the probability if the standard deviation were increased to 3%?
5. How can we apply this process to find the probability of earning more than a certain gain?

Tip: You can use a Z-table or a calculator to find the cumulative probability for any Z-score when dealing with normal distributions.