Central Limit Theorem
What does the Central Limit Theorem say about the sampling distribution of the sample mean?
As the sample size increases, the sampling distribution of the sample mean becomes approximately normal.
What condition allows the use of a normal model when the population distribution is not normal?
The sample size must be at least 30.
What is a sampling distribution?
The distribution of a statistic from all possible samples of a given size.
Which statistic is used for quantitative data?
Sample mean.
What happens to σₓ̄ as the sample size increases?
It decreases.
What happens to the shape of the sampling distribution of the sample mean as the sample size increases?
It becomes more nearly normal.
Why must samples be random when creating sampling distributions?
To ensure the sample is representative of the population.
What happens to variability as sample size increases?
It decreases.
Which statistic is used for success/failure data?
Sample proportion.
What is the formula for σₓ̄?
σ / √n
A population has μ = 60 and σ = 12. Find the standard deviation of the sampling distribution of the sample mean when n = 36.
σₓ̄ = 12 / √36 = 2
A population is strongly skewed and n = 12. Can a normal model be used?
No
σ = 10 and n = 25. Find σₓ̄.
10 / √25 = 2
110 out of 200 students prefer online learning. Find p̂.
p̂ = 0.55
σ = 18 and n = 9. Find σₓ̄.
6
A population has μ = 100 and σ = 20. If n = 50, find P( x̄ > 105 ).
σₓ̄ = 20 / √50 ≈ 2.83
z = (105 − 100) / 2.83 ≈ 1.77
P ≈ 0.038
A population distribution is unknown, and n = 45. Can a normal model be used?
Yes
μ = 80, σ = 12, n = 16. Find P(78 < x̄ < 82).
σₓ̄ = 3, z = ±0.67, P ≈ 0.75
p = 0.60 and n = 100. Find the standard deviation of p̂.
√[0.6(0.4)/100] ≈ 0.049
How does variability change when the sample size decreases from 100 to 25?
It doubles.
A population has σ = 15. What minimum sample size is needed so that σₓ̄ < 1?
15 / √n < 1 → n > 225 → 226
A: z = ±1.5 → P ≈ 0.866
The sampling distribution of x̄ has μ = 50 and σₓ̄ = 2. Find P(47 < x̄ < 53).
z = ±1.5 → P ≈ 0.866
μ = 90, σ = 15, n = 25. Find P(x̄ < 85).
z ≈ −1.67, P ≈ 0.05
p = 0.40 and n = 200. Find P(p̂ > 0.45).
z ≈ 1.43, P ≈ 0.076
What sample size is required so that σₓ̄ ≤ 2 when σ = 20?
n ≥ 100