CLT and Proportions

I would like to ask those of you who know their CLTs and Proportions a few questions.
I am completing my HW right now and wanted do ask you guys a few Qs please. :wave:

Example: The mean price of shoes in a certain store is $585 with a standard deviation of $195. I am randomly selecting 35 pairs of shoes in this store. What is the probability that the mean price will be between $500 and $690 ?

I identified this example as CLT.

I started with the x bar being $585, $195 and n=35. Then wrote down that the mean of my x bar is basically the mean = $585, then calculated the sigma for the x bar and wrote down my probability as P ($500< x bar < $690). then calculated the z score for the $500 and $690 and calculated that my area on diagram based on the z scores from the table. I came up with 99% probability that my shoe prices will fall between $500 and $690.

Does this sound any good? I would appreciate feedback.