Chapter 7
Key Ideas
Confidence Interval, Confidence Level
Point Estimate, Margin of Error, Critical Value, Standard Error
t distribution, Chi-Square distribution
Section 7-1: Overview
All of the material in Chapters 4-6 forms a foundation of what is called
inferential statistics
.
We already dealt with inferential
statistics in Chapter 10 in a regression setting.
Now, we will explore estimation.
Here is an outline of the main idea of inferential statistics:
1.
There is a population of interest (i.e. the group we want to know something about)
2.
We draw a random sample of size
n
from the population.
3.
We compute statistics from the sample,
4.
We use these statistics to
estimate
similar parameters in the population.
For example, suppose we want to know what percentage of Denison students own a red car.
To estimate this, we take a random
sample of 100 students and find out what percentage of those 100 students own red cars.
Then, we say that the sample percentage
should to close to the actual percentage of
all
Denison students who own red cars.
In this example:
Population of Interest: All Denison Students
Sample Size:
n
= 100
Statistic from the Sample: The percentage of students sampled who own red cars
Parameter in the Population: The percentage of all Denison students who own red cars
How do we actually estimate the parameter, though?
General Estimation Framework
Suppose we want to estimate a parameter (e.g. population proportion, population average, etc.).
The first thing to notice is that it would be impossible to exactly pinpoint the value with 100% accuracy without sampling every single
member of the population, since there would always be some uncertainty.
As a result, the best we can do is make a guess at the true
value, and then include a margin of error based on a certain level of confidence we have in our results.
The estimate and the margin of
error form something called a
confidence interval
.
A confidence interval is made of 2 different parts.
1.
The
point estimate
is the sample statistic (this is our best guess at the true parameter value given our sample).
2.
The
margin of error
is added and subtracted from the point estimate to make the interval.
It can also be subdivided into two parts:
a.
A
critical value
from a distribution (more to come on this later)
b.
The
standard error
of the point estimate (more to come on this as well)
The confidence interval (CI) has this form:
CI = (Point Estimate) ± (Margin of Error)
= (Point Estimate) ± (Critical Value)*(Standard Error)
Of course there is no guarantee that the true population parameter will be in this interval, so we have to make some sort of statement
about the chances that this will be true.
The
confidence level
is the probability that the interval actually covers the true population parameter.
Often, the confidence level is
denoted (1 –
α), where α is the chance that it does
not
cover the true parameter.
For example, if α = 0.05, then the confidence level is
0.95, or 95%.
Thus we would say that we are 95% confident that the interval covers the parameter.