Parameter
A number that describes some characteristic about a population
Statistic
A number that describes some characteristic about a sample
Notations of Statistics
x̄ - Sample mean
Sx - Sample SD
P^ - Sample proportion
Notations of parameters
M - Pop mean
σ - Pop SD
p - Pop proportion
AP Exam Tip for parameters
Students often lose credit when naming a parameter, try to use words like “true” or “all”
Sampling variability
The idea that different random samples from same pop, produce different stat values each time
Sampling distribution
The dist of stat values taken from ALL POSSIBLE random samples of the same size from same pop
AP Exam Tip for Sampling dist
Be specific when you identify what dist you are talking about
Unbiased estimator
A statistic used to estimate an unknown Parameter, unbiased if the mean of its sampling dist equals the value of the parameter we are estimating
Bias and variability
Accuracy:
To get a trustworthy estimate of an unknown parameter, start by using an unbiased estimator
Precise:
Larger sample sizes help reduce spread (variability)
In the sampling dist of statistics
AP Exam Tip for precision & accuracy
Sample sizes have to do with precision NOT accuracy
Sampling dist of sample proportion p^
Describes the dist of all p^ values from all possible samples, of a certain size from same pop
Shape of sampling dist p^
Depends on n and p
Skewed right
n= small
p= close to zero
Skewed left
n= small
p= close to 1
More symmetric
n= larger
p= close to 0.5
Center of sampling dist p^
Because p^ is an unbiased estimator of P, the mean of its sampling dist is equal to p
Mp^ = P
Variability
Larger sample sizes reduce variability
SDp^ = square root of p(1-p) / n
SD of the sampling dist of p^
Important facts about sampling dist of p^
Shape: Sampling dist of p^ is Approx normal long as
np >_ 10 and n(1-p) >_ 10
(Large counts condition)
Center: Mp^ = P(unbiased estimator)
Spread: SDp^ = square root p(1-p) / n
(Check 10% condition)
Using normal approximation for p^
When large counts condition
Passes we can use Normcdf to calculate probabilities
The sampling dist of the sample mean x bar
Describes the dist of values taken by sample mean (x bar), in ALL possible samples of a certain size, from same pop
The sampling dist of x bar: mean and SD
Mx bar = M (unbiased estimator)
Mx bar is center of sampling dist
M is pop mean
Check 10% condition
SD of x bar = SD over square root n
SD of x bar is spread of sampling dist
AP Exam Tip Notation
Notations matters
X bar, M, M x bar, SD, SD x bar
USE THESE CORRECTLY
Sampling from a normal pop: Shape
If pop is normally distributed, sampling dist of x bar is also normally distributed with
M of x bar = M
SD of x bar = SD over square root n
Doesn’t matter the sample size!
The sampling dist of the sample mean x bar when sampling from a normal pop
Center
M of x bar = M
Spread
SD of x bar = SD over square root n
If pop ND so is the sample dist shape
Central Limit Theorem (CLT)
When drawing an SRS from ANY pop If n is large, the sampling dist of x bar is approx normal
Shape of the sampling dist of the sample mean x bar (norm vs not norm)
Pop normal:
Sampling dist of x bar is norm no matter what sample size is
Pop not normal:
Sampling dist of x bar is normal if n >_ 30 (CLT)
ONLY APPLIES IF SAMPLE MEANS NOT PROPORTIONS
