Probability Distribution from a Existing Distribution?

I'm a little confused by this problem which seems simple, but I'm not really sure how to solve it.

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It says: The number of hamburgers ordered in a day is normally distributed with a Mean of 45 and Standard Deviation of 8.4, Then asks:

What is the probability Distribution (type of distribution, mean and Standard Deviation) of the average hamburgers ordered for 50, 100, 1000 Customers?
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Would the type of distribution just be normal for all 3 sizes, or is there a way to calculate the type of distribution?
Our book basically says if your n > than like 30 I think it can be a normal distrib, or 50 if theres some bad out liers

I'm not really sure how to calculate the mean, and if I even need to for a already given mean and SD, when you are just given what I guess is the sample size. I have a formula to calculate the Standard deviation of the Sample Mean which is:

Standard Deviation / Square root of the sample size

I think this is right,
for 50: 8.4/[SQRT(50)] = SD of 1.188
for 100: 8.4/[SQRT(100)] = SD of .84
for 1000: 8.4/[SQRT(1000)] = SD of .266


But I'm not sure if these are the correct numbers for the SD, and how to calculate the mean, or maybe I just use the original mean of 45 given in the problem?

Thanks for any help.

The average number of hamburgers ordered by \(\displaystyle n\) customers on a given day has a normal distribution with a mean of 45 and a standard deviation of \(\displaystyle 8.4/\sqrt{n}\). That is what this problem is trying to say.

When the original distribution is normal, then the distribution of averages is also normal. This is true for every sample size. The mean does not change. The standard deviation is reduced by the square root of the number of things that went into the average.

The guidelines in textbooks that talk about sample sizes greater than 30 and so on are just that: guidelines. In other words, if the original distribution is not normal but it isn't too "far" from normal, then the distribution of averages, or means, from "large" sample sizes "look" like they are close to normal distributions.