Point Estimator
Statistic that provides an estimate of a population parameter.
Point Estimate
Our "best guess" at the value of an unknown parameter.
Confidence Interval
Interval calculated from the data for a parameter.
estimate +/- (Critical Value) x (Standard Deviation of Statistic)
Confidence Interval Equation
Margin of Error
Tells how close the estimate tends to be to the unknown parameter in repeated random sampling.
Confidence Level C
Gives the overall success rate of the method for calculating the confidence interval. That is, in C% of all possible samples, the method would yield an interval that captures the true parameter value. "We are __% confident that the interval captures the a
Confidence Level
95% of all possible samples of a given size from this population will result in an interval that captures the unknown parameter.
Critical Value
Depends on both the confidence level C and the sampling distribution of the statistic.
(Critical Value) x (Standard Deviation of Statistic)
Margin of Error Equation
Larger Samples (Sample Size n Increases)
What gives more precise estimates?
Smaller % and Narrower Interval
Less confident = ___ & ___
Random; Normal; Independent
To use a confidence interval, the situation must meet what conditions?
Random
Comes from a random sample or SRS.
Normal
n > 30 or np > 10 AND n(1-p) > 10
Independent
n< 1/10 N (Sampling more than 10% of the population)
.10/2 =.05 ? Z.05 = 1.645
Z score for 90% Confidence
.05/2 = .0250 ? Z.025 = 1.96
Z score for 95% Confidence
.025/2 = .0125 ? Z.0.0125 = 2.326
Z score for 98% Confidence
Standard Deviation (Sigma)
Sigma of p hat = Square Root of ((p(1-p))/n)
Standard Error
When the standard deviation of a statistic is estimated from the data, the result is called ___.
One-Sample z Interval for p
p hat +/- (critical value) x (Square Root of ((p(1-p))/n))
State, Plan, Do, Conclude
Four-Step Process
State
What parameter do you want to estimate and at what confidence level?
Plan
Identify the appropriate inference method and check conditions.
Do
If the conditions are met, perform calculations.
Conclude
Interpret your interval in the context of the problem.
One-Sample z Interval for a Population Mean
x bar +/- (Critical Value) x (sigma/n) (Use when sigma is known)
z= (x bar - mu)/(sigma/square root of n) ? mu = 0 and sigma = 1
When the sampling distribution of x bar is close to Normal, we can find probabilities involving x bar by standardizing:
When we don't know sigma
When do we use the following formula: ?? = (x bar - mu)/(s of x / square root of n)
t = (x bar - mu)/(s of x/square root of n)
T-Distribution
Degrees of Freedom
df = n-1
Standard Error of the Sample Mean
x bar = (sample standard deviation or s of x) / square root of n --- It describes how far x bar will be from mu, on average, in repeated SRS of size n.
One-Sample t Interval for a Population Mean
x bar +/- t of (n-1) x (s of x / square root of n)
Robust Procedures
An inference procedure is called ____ if the probability calculations involved in that procedure remain fairly accurate when a condition for using the procedure is violated.
Sample Sizes LESS than 15
Use t procedures if the data appear close to Normal (symmetric, single peak, no outliers)
Sample Sizes AT LEAST 15
The t procedures can be used except in the presence of outliers or strong skewness.
Large Samples
The t procedures can be used even for clearly skewed distributions when the sample is large, roughly n > 30.