Confused about sampling distribution/central limit theorem

[Herondaleheir]

So I'm trying to study for a test and I'm stuck on two textbook questions. I'm having trouble grasping the concept of sampling distributions and when to apply certain rules/when to categorize a population as normal.

I have some points in the testbook and I'm trying to apply this knowledge to these questions but I'm just confused. This post is kind of messy, so I apologize about that but I would appreciate some help. I bolded my attempted answers.

Textbook states:
Central Limit Theorem – regardless of the shape of the population from which a sample was drawn, the sampling distribution of the mean of the sample will have a mean µ (which is equal to the population from which we’re sampling) and a standard deviation σ equal to σ/√n.

When n is large enough (n ≥ 30), the sampling distribution of the sample mean x̄ is approximately normally distributed (roughly mound shaped). And thus, has mean µ and standard deviation σ/√?

1. Random samples of size n were selected from populations with the means and variances given here. Find the mean and standard deviation of the sampling distribution of the sample mean in each case:

a. n = 36, μ = 10, σ^2 = 9

So I'm assuming because n is greater than 30, I can do μ = 10, and σ = 3/√36 = 1/2 which is correct in the textbook's answers.

b. n = 100, μ = 5, σ^2 = 4

Since n ≥ 30, μ = 5 and σ = 1/5 which is also right.

c. n = 8, μ = 120, σ^2 = 1

But over here, since n is not greater or equal to 30, I don't really know what to do. The textbook says μ = 8 and σ = 1/√8. I don't know why the mean isn't 120 like the others? What rule is it following if it's not the central limit theorem?

And then I'm just confused in general how to get the answers for 2, as it's not given in the textbook. I don't really understand how to determine if a sampled population is normal/not normal and how that changes the sampling distribution.

2. Refer to exercise 1.
a. If the sampled populations are normal, what is the sampling distribution of x̄ for parts a, b, and c?
b. According to the Central Limit Theorem, if the sampled populations are not normal, what can be said about the sampling distribution of x̄ for parts a, b, and c?

 

[Harry_the_cat]

I'd say the book is wrong for part (c) - the mean should be 120 as you said.

The value of n doesn't affect the mean as stated by the CLT – "regardless of the shape of the population from which a sample was drawn, the sampling distribution of the mean of the sample will have a mean µ (which is equal to the population from which we’re sampling) and a standard deviation σ equal to σ/√n ".

The size of n affects the shape of the sampling distribution: "When n is large enough (n ≥ 30), the sampling distribution of the sample mean x̄ is approximately normally distributed (roughly mound shaped). And thus, has mean µ and standard deviation σ/√?." This is true regardless of the shape of the population distribution.

Basically,
(1) if the original population is normally distributed, then the sampling distribution will be normally distributed (doesn't matter what n is)
(2) if the original population is NOT normal AND n>=30, then the sampling distribution will be approximately normally distributed
(3) if the original population is NOT normal and n<30, then you can't assume that the sampling distribution will be normal or not

Hope that helps you answer Q2.