Point Estimator & Point Estimate
Point estimator — the name of a statistic that provides an estimate of a pop parameter
Point estimate — the actual value is called the point estimate
Confidence Interval
An interval of plausible values for a pop parameter based on a sample data
Point estimate + or - Margun of error
Confidence level and how to interpret
Gives the overall success rate of the method used to calculate the confidence interval
Interpret:
“In __ % of All possible samples, the interval calculated from the sample data will capture the true parameter value”
How to interpret a confidence interval
“We are __ % confident that the interval from __ to __ captures the true [ parameter in context ]”
Describes the PARAMETER NOT the STATISTIC
Margin of error
Describes how far at most we expect our point estimate to vary from the true parameter value
Interpreting a confidence level
In __ % of all possible samples the interval calculated from the point estimate will capture the true parameter value
SEE PAGE 4 for examples
AP Exam Tip: Interval vs Level
The INTERVAL is the actual set of plausible values
The LEVEL describes the overall capture rate
C-Level IS NOT a probability
Affecting the Margin of Error
In general we want small M.O.E. (Margin of error) since we get a more precise estimate
MOE gets smaller when:
- C-Level decreases
- Sample size increases
Margin of error DOES NOT account for
A poor sampling method (ex. Convenience sampling)
Critical Value
The number of SDs wide you need to make your interval to match C-level
One sample z interval for a population proportion
p^ +_ z* ( square root p^(1-p^) / n
Standard Error of p
z* —> critical value
Conditions for Estimating p
1) Data come from a random sample
2) Large Counts
np^ >_ 10 n(1-p^) >_ 10
3) 10% condition
N >_ 10(n)
Conditions for constructing a confidence interval about a proportion
Always check 3 conditions
SEE PAGE 6 for examples
Calculating Critical values (z*)
Use InvNorm
See page 7 for examples and practice
Confidence Intervals: 4 step process
1) State parameter of interest
2) Check conditions
3) Calculate critical value and Interval
4) Intepret in context
SEE PAGE 8 for examples
Sample size for desired margin of error when estimating p
What value to use for p^
MOE > z* square root of p^ (1-p^) / n
Solving for n
1) Use value from previous study
2) Use p^ = 0.5
Confidence intervals for Quantitive data: Sampling dist of a sample mean x bar
Normal condition from central limit theorem (CLT)
Sample size n>_ 30
Mean of sampling dist is same as mean of pop
SDx bar = SD / square root n
CI = Point estimate + or - MOE
SEE PAGE 11
One sample z interval for pop mean
The confidence interval is x bar + or - z* (SD / square root n)
This formula can only be used when normal condition is met (CLT) and independence condition (10%) is met
SEE PAGE 11
Choosing sample size for pop mean
Quantitative data
MOE >_ z * (SD / square root n)
Find critical value z * from standard normal curve Table or InvNorm
Has to be below certain margin of error
Choosing sample size for pop mean
Quantitative data
MOE >_ z * (SD / square root n)
Find critical value z * from standard normal curve Table or InvNorm
Has to be below certain margin of error
Used for QUANTATATIVE data
Formula for sample size with QUANTITATIVE data
n >_ (z* SD / MOE) squared
When we don’t know SD we use Sx instead
t distributions
When M and SD we can use z-score formula, SD of sampling mean along with a substitution of Sx instead of SD to get a t distribution
Z score formula = value - mean / SD becomes
Statistic - parameter / SD of statistic
This distribution is NOT a normal dist it has larger tails and is called a t distribution
T distributions vs Normal dists
T distribution is still:
Bell shaped
Unimodal
Symmetric
There is a different t dist for each ____ . We specify each t dist by ______
The larger spread comes from substitution of __ for SD which introduces more _____
As degrees of freedom _______ the t density curve approaches the ________ more closely
Sample size
Degrees of freedom (df)
df = n-1
Sx
Variability
Increase
Normal dist
SEE PAGE 13
