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. I attempted 1 but would like clarification, and I don't know how to do 2.
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/√n.
- 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 apply the central limit theorem and do μ = 10, and σ = 3/√36 = 1/2 which is the correct answer in the textbook.
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'm guessing I can't use central limit theorem and don't really know what to do. The textbook says μ = 8 and σ = 1/√8. I don't know why the same standard deviation formula is used but the mean isn't the original value of μ like the others? Why 8? 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.
- 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?