Basic Terms
What is a sampling distribution?
The distribution of a statistic (like the mean) from all possible samples of a given size
What is μₓ̄ equal to?
μ (the population mean)
What does the CLT tell us about the sampling distribution of the mean as n increases?
It becomes approximately normal regardless of population shape
If μ = 71.2 and σ = 16.95 with n = 5, what happens to σₓ̄ compared to σ?
It’s smaller
How do you find a sampling distribution mean in StatCrunch?
Compute sample means using Stat → Summary Stats
What is a statistic in this context?
A value (like the sample mean) calculated from sample data
What happens to σₓ̄ when the sample size increases?
It decreases
Why is the CLT important in statistics?
It allows us to use normal probability methods even when the population isn’t normal
If n increases from 5 to 30, what happens to σₓ̄?
It decreases significantly
What menu can help simulate multiple samples in StatCrunch?
Data → Simulate or resample
What does μₓ̄ represent?
The mean of the sampling distribution
What does the shape of the sampling distribution look like if the population is normal?
Normal
What is the minimum recommended sample size for CLT to apply for non-normal populations?
n > 30
What is the mean of the means from repeated samples equal to?
The population mean
How can StatCrunch visually show the CLT?
By graphing sampling distributions of sample means
How does σₓ̄ compare to σ?
σₓ̄ is smaller than σ
What happens to the sampling distribution when the population is not normal but n > 30?
It becomes approximately normal
If the original data is normal, what is the shape of the sampling distribution of the mean?
Normal
Why does the sampling distribution get narrower with larger n?
Because variability in sample means decreases
If you double n in StatCrunch simulations, what should happen to the histogram?
It becomes more centered and less spread
What does the Law of Large Numbers imply about sample means?
As n increases, sample means get closer to the population mean
What is the formula for σₓ̄?
σ / √n
What happens to variability as sample size increases?
It decreases (the mean stabilizes)
If the original population is skewed but n is 40, what shape will the sampling distribution of the mean have?
Approximately normal
What shape will the histogram approach with large n?
A normal bell curve