The question I'm trying to answer says that the mean weight of luggage checked by a randomly selected passenger is 40 lb, and standard deviation is 10 lb. Luggage weights are independent. What is the probability that the average weight of a random sample of 36 pieces of luggage is at most 35 lb?
Since the sample size is 36, the distribution is normal according to CLT.
Past that I'm stuck. I know that its possible to transform it to become a standard normal distribution, but I'm struggling to do that.
Hint: Consider what the average weight of a piece of luggage being 35 lbs means about the weight of the whole set of luggage.
Once you get that, consider $Z = \sum_{i=1}^{36}w_i$ (where $w_i$ is the weight of the $i^{th}$ suitcase) as a random variable defined as a sum of random independent, identically distributed variables. That is, if you add a bunch of random variables with the same distribution, what is the distribution of their sum? That should be enough to get you started, I hope.