Distributions and means - confidence interval question

#1
Hi folks, :wave:

I think I did my research but I would like to be sure before continuing.

My task is to find the mean price of a given item in a given country and I need to know how accurate my sample mean is vs. the actual population mean.

Here is what I have. (My proposed, probably flawed, methodology)

1. My population consists of prices of good X (homogeneous and perfectly fungible).
2. My sample has to contain at least 30 objects in order for the t-dist to approach the normal dist.
3. I can use a confidence interval to test how accurate my mean is AND I can adjust the number of objects in my sample in order to obtain a more or less accurate confidence interval.
4. This means I can now divide the sample mean by any given price of good X of a domestic producer and create an index.

My doubts are as follows:

a. I cant just assume a t or normal distribution right? I mean wouldn't I need to do a scatter plot or something to get an idea of the dist. and then maybe some other tests to be sure of the dist.?
b. Are there any alternative (better) ways to measure the accuracy of my mean?

I must admit I am quite embarrassed I took courses in statistics, probability, econometrics and advanced quantitative methods in my economics undergrad and don't remember anything... please be so kind as to help my Swiss cheese brain with this. Thanks.

Fresh