Sampling Distribution Basics
What is a sampling distribution?
The distribution of a statistic (like the mean or proportion) over all possible samples of a given size from a population
What is the mean of the sampling distribution of the sample proportion p^?
p, the population proportion
What is the mean of the sampling distribution of the sample mean xˉ?
μ, the population mean
What does the Central Limit Theorem (CLT) state?
The sampling distribution of the sample mean becomes approximately normal as the sample size increases
You sample 50 people from a population with p = 0.6. What's the expected value of p^?
0.6
What is the center (mean) of a sampling distribution of a sample mean?
The population mean, μ
What is the standard deviation of the sampling distribution of p^?
Square root of (p(1-p))/n, assuming conditions are met
What is the standard deviation of xˉ when sampling from a population with standard deviation σ?
σ/(square root of n)
What sample size is typically considered large enough for the CLT to apply?
30 or more
A population is skewed right. You take a random sample of size 40. Can you use a normal model to describe the sampling distribution of the sample mean? Why or why not?
Yes, because the sample size is large enough (n ≥ 30) for the Central Limit Theorem to apply.
What does the Law of Large Numbers say about sample means as sample size increases?
They get closer to the population mean.
What two conditions must be met to use the normal model for p^?
1) Random sample
2) np≥10 np≥10 and n( 1 − p)≥10 n(1−p)≥10
Under what conditions is the sampling distribution of the sample mean approximately normal?
If the population is normal or n≥30
Why is the CLT important in statistics?
It allows us to use normal probability formulas even when the population distribution is not normal.
In a random sample of 100 people, 60% say they like pineapple on pizza. Assuming the population proportion is 0.6, what is the standard deviation of the sampling distribution of p^?
Square root of (0.6)(0.4)/100 = 0.049
How does increasing the sample size affect the variability of a sampling distribution?
It decreases the variability.
If a population proportion is 0.4 and you take a random sample of size 100, what is the standard deviation of p^?
Square root of (0.4)(0.6)/100 = 0.049
A population has a mean of 50 and a standard deviation of 10. What is the standard deviation of the sample mean for samples of size 25?
10/(square root of 25) = 2
For a population with mean 100 and standard deviation 20, what would be the approximate shape of the sampling distribution of the mean for n=40?
Approximately normal. (40≥30, CLT suggests approximately normal)
In a quality check, a sample of 64 batteries has an average charge of 98 hours. Population mean is 100, SD is 8. What's the probability of getting a sample mean this low or lower?
z = (98−100)/(8/square root of 64) = −2
P(Z < -2) ≈ 0.0228
What condition must be met for the sampling distribution of the sample mean to be approximately normal when the population is not normal?
The sample size must be large (n≥30, CLT)
Why can we say the sampling distribution of p^ is approximately normal even if the population distribution is not?
The Central Limit Theorem
What is the shape of the sampling distribution of xˉ if the population is heavily skewed and n=10 ?
It is likely skewed and not approximately normal (sample size is too small for CLT)
A population is right-skewed with mean 60 and standard deviation 12. What can we say about the distribution of the sample mean for samples of size 50?
Approximately normal. (50≥30, CLT suggests approximately normal)
A population has a mean of 75 and a standard deviation of 20. What is the probability that a sample of size 36 has a mean greater than 80?
σxˉ = 20/square root of 36 = 3.33
z = (80-75)/3.33 ≈ 1.5
P(xˉ>80) = P(Z>1.5) ≈ 0.0668