Welcome to the introduction
to confidence intervals.
Suppose you are trying to determine the mean or average rent
of a two bedroom apartment in your city.
You might look online, write down several rents listed
and find the average.
You would have obtained a point estimate of the true mean.
For example, you might have found an average rent
of $1,200 per month.
We often use sample data to make generalizations
of an unknown population.
This part of statistics is called inferential statistics.
The sample data help us to make an estimate
of a population parameter.
We realize that the point estimate
is most likely not the exact value
of the population parameter but close to it to be useful.
After calculating point estimates
we construct interval estimates called confidence intervals.
So for example, for the rent, the interval estimate
might be from 1,000 to $1,400 per month.
If you worked in a marketing department
of an entertainment company, you might be interested
in the mean number of songs
a consumer downloads a month from iTunes.
If so you could conduct a survey
and calculate the sample mean, x bar,
and the sample standard deviation s.
You would use x bar to estimate the population mean,
and s to estimate the population standard deviation.
The sample mean x bar is the point estimate
of the population mean mu.
The sample standard deviation s is the point estimate
for the population standard deviation sigma.
Each x bar and s is called a statistic.
A confidence interval is another type of estimate,
but instead of being just one number,
it is an interval of numbers.
It provides a range of reasonable values
in which we expect the population parameter to fall.
There is no guarantee that a given confidence interval
does capture the parameter
but there is a predictable probability of success.
Suppose for the iTunes example
we do not know the population mean mu
but we do know the population standard deviation sigma
is equal to one, and our sample size is 100.
Then by the central limit theorem, the standard deviation
for the sample mean is equal to the standard deviation
of the population divided by the square root
of the sample size, which is equal to one divided
by the square of 100, which equals 1/10 or 0.1.
The empirical rule which applies
to bell-shaped distributions says,
that in approximately 95% of the samples,
the sample mean x bar will be
within two standard deviations
of the population mean mu, for the iTunes example
two standard deviations is equal to two times 0.1 or 0.2.
The sample mean x bar is likely to be
within 0.2 units of mu.
In other words, mu is between x bar minus 0.2,
and x bar plus 0.2 in 95% of all of the samples.
Meaning the 95% confidence interval
is from x bar minus 0.2 to x bar plus 0.2.
Again, for the iTunes example,
suppose that a sample produced a sample mean
of x bar equals two, then the 95% confidence interval
for the unknown population mean mu
is between two minus 0.2 and two plus 0.2,
which means 95% confidence interval is 1.8 to 2.2.
We say that we are 95% confident
that the unknown population mean number of songs downloaded
from iTunes per month is between 1.8 and 2.2.
The 95% confidence interval implies two possibilities.
Either the interval from 1.8 to 2.2
contains the true mean mu,
or our sample produced an x bar that is not
within 0.2 units of the true mean mu.
The second possibility happens
for only 5% of all of the samples.
100% minus 95%.
Remember that a confidence interval is created
for an unknown population parameter,
like the population mean mu.
The confidence intervals for some parameters
have the following form.
The point estimate minus the margin of error comma,
the point estimate plus the margin of error.
The margin of error depends
on the confidence level,
or percentage of confidence and the standard error
of the mean.
When you read newspapers and journals,
some reports will use the phrase margin of error,
other reports will not use that phrase
but include a confidence interval
as the point estimate plus or minus the margin of error.
These are two ways of expressing the same concept.
We will begin to calculate confidence intervals
in the next lesson.
I hope you found this helpful.