Yes, it is appropriate to use the normal distribution to find probabilities for the sample mean $\bar{x}$ in this case.

Here's why: According to the Central Limit Theorem, if we draw a sample of size $n \geq 30$ from a population with any distribution, the sampling distribution of the sample mean $\bar{x}$ will be approximately normal. Here, the sample size is 37, which is greater than 30, so we can assume that the distribution of the sample mean is approximately normal.

Would you like a more detailed explanation, or do you have any specific questions?

Here are 5 additional questions to deepen understanding:

1. How does the Central Limit Theorem justify the use of a normal distribution for larger sample sizes?
2. What would happen if the sample size were significantly smaller, say $n = 10$?
3. How do you calculate the standard error of the mean for a sample?
4. Why is it useful to approximate the distribution of sample means as normal?
5. What is the difference between the population distribution and the sampling distribution?

Tip: For sample sizes larger than 30, the Central Limit Theorem usually allows us to assume normality in the sampling distribution of the mean, making it easier to calculate probabilities and confidence intervals.