Sampling-Dist. Basics
This term describes the distribution of a statistic over all possible random samples.
What is a sampling distribution?
Symbol used for a sample proportion.
p^hat
Symbol for the population mean.
μ
CLT tells us the sampling distribution of xˉbar becomes roughly what shape for large n
Normal
High variability shows up on a sampling-dist. graph as a wide or narrow curve?
wide
Statistic vs. parameter: Which one is unknown but fixed for a population?
Parameter
Mean (center) of the sampling distribution of p^hat
p
Formula for SD of xˉbar when population SD is σ
σ/sqrt(n)
Minimum “rule-of-thumb” sample size that often makes CLT safe for xˉbar when the population is not extreme.
30
educing sample size will do what to the SD of a statistic— increase or decrease it
increase
A “good estimator” must have two desirable features: it should hit the target on average and show little scatter. Name them.
non-bias, low variability
Formula for the SD of p^hat
sqrt(p*(1-p)/n)
If a population is strongly skewed, what must be true about n to trust the CLT for xˉbar
n >= 30
Explain in one phrase why the CLT is powerful for statisticians.
It lets us use Normal probabilities even when the population isn’t Normal.
Bias affects the ______ of a sampling distribution; variability affects the ______
Center; spread
The long-run guarantee that a statistic’s average value will get closer and closer to the population parameter as the number of samples grows is called the ______.
Law of Large Numbers
In Binomial(40, 0.25), what is the mean of p^hat
0.25
Population SD = 15. For n=25, give SD of xˉbar
15/5=3
true/False: If the population itself is Normal, CLT is unnecessary—xˉbar is Normal for any n
true
Choose the better estimator (lower variability): n=50 or n=200 random sample?
n=200
In plain language, what does the standard deviation of a sampling distribution tell you?
the typical distance between the statistic and the true parameter
You want the standard deviation of p^\hat{p}p^ to be one-third of its current value. By what factor must you multiply the sample size nnn?
9
Explain why doubling nnn does not cut SD of xˉbar in half
SD falls by 1/sqrt{n}; doubling n multiplies SD by 1/sqrt{2}, not ½
Give a real-world example where CLT justifies using a Normal model for the average delivery time of a company.
Any reasonable example “UPS averages 50 packages daily; by CLT the mean daily delivery time ≈ Normal.”
If a method has no bias but huge variability, describe how the histogram of its sampling distribution will look relative to the parameter.
Centered at the parameter but very wide.