14 - Sampling, Estimates and Resampling

Question 1

Define the Sample Mean Distribution.

Answer 1

𝑋̅ ~𝑁 (𝜇, 𝜎²/𝑛) where 𝑋̅ is the sample mean distribution, 𝜇 is the population mean, 𝜎² is the population variance, and n is the sample size.

Question 2

What is the Standard Error?

Answer 2

The standard deviation of the Sample Mean Distribution, calculated as 𝜎/√𝑛.

Question 3

State the Central Limit Theorem (CLT).

Answer 3

If a random sample of size n is taken from any distribution with a mean, 𝜇, and standard deviation, 𝜎, then 𝑋̅ will have mean, 𝜇, and standard error, 𝜎/√𝑛. This distribution will be approximately normally distributed given that 𝑛 is sufficiently large (𝑛 ≥ 30).

Question 4

When can you assume the Sample Mean Distribution is normally distributed, even if the underlying data isn’t?

Answer 4

When the sample size, n, is sufficiently large (generally, n ≥ 30), according to the Central Limit Theorem.

Question 5

Why is the Central Limit Theorem important in hypothesis testing?

Answer 5

It allows us to perform hypothesis tests on sample means without needing to know the underlying distribution of the population, as long as the sample size is large enough.

Question 6

In hypothesis testing with sample means, what happens to the distribution of the sample mean if the original population is normally distributed?

Answer 6

The Sample Mean Distribution is normally distributed for any sample size, n.

Question 7

What formula do you use for a test statistic when performing a hypothesis test about a population mean, using the sample mean distribution? (Assume you know the population standard deviation)

Answer 7

z = (𝑥̅ - 𝜇) / (𝜎/√𝑛)

Question 8

What should you consider when evaluating a hypothesis test that uses the Central Limit Theorem?

Answer 8

Whether the sample was random, and if the sample size is large enough (n ≥ 30) for the CLT to apply. If the sample isn’t random, the results of the test may be invalid.