04 Sampling Distributions

Question 1

What is a sampling distribution?

Answer 1

The distribution of a statistic (such as the mean, standard deviation, proportion) calculated from all possible samples of a fixed size.

It tells you the number of possible samples which have a certain value for a statistic.

Question 2

What do μ, σ, and p represent?

Answer 2

The mean, standard deviation and proportion of the whole population the sample is drawn from

Question 3

What do x̄, s and p̂ represent?

Answer 3

The mean, standard deviation and proportion of a sample taken from the larger population.

Question 4

How can a sample statistic (eg x̄, s²) provide an estimate of the population parameter?

Answer 4

We can calculate the mean of the statistic; eg the mean value of the x̄’s from all the different samples

Question 5

If the mean of the sample statistic is equal to the true population parameter, the statistic is said to be…?

Answer 5

Unbiased

Question 6

Which of x̄, s, s² and p̂ are unbiased estimates of the population parameters?

Answer 6

x̄, s² and p̂

Question 7

What does the central limit theorem say?

Answer 7

If an SRS of size n is taken from a population, then x̄ and p̂ follow approximately normal distributions

• x̄ ~ N(μ, σ/√n)
• p̂ ~ N(p, √[p(1-p)/n])

Question 8

When can the central limit theorem be applied?

Answer 8

• When np > 10 and np(1-p) > 10
• When the population size N > 10n
• When n > 30

Question 9

If we have two samples, what do we know about the distribution of the differences between the two sample’s means and proportions?

Answer 9

• x̄₁-x̄₂ ~ N(μ₁-μ₂, √[σ²₁/n₁-σ²₂/n₂])
• (p̂₁-p̂₂) ~ N(p₁-p₂, √(p₁(1-p₁)/n₁ + p₂(1-p₂)/n₂)
• If np and n(1-p) > 10
• N > 10n

Question 10

If we have an SRS of size n taken from a population with mean μ and standard deviation σ, what can we say about the distributions of x̄ and p̂?

Answer 10

• x̄ ~ N(μ, σ/√n)
• p̂ ~ N(p, √[p(1-p)/n]