Confidence Intervals- CHEAT SHEET
WHAT IS A CONFIDENCE INTERVAL?

A confidence interval for the unknown population mean μ is an interval (or range) of
plausible values for μ. It is constructed such that with a chosen degree (or level) of
confidence C, the value of the unknown population mean will be captured inside the
interval.

FOR A POPULATION WE KNOW THAT X SHOULD FALL BETWEEN:
WHAT DOES C STAND FOR?

C provides information on how much “confidence” we can have in the method used
to construct the CI


C usual choices are: 90%, 95%, and 99%
C can be interpreted as the rate of success for the method used to construct CI in the
long run

C is the % we will use to get our z-score for the following formula:
WHAT ARE THE CORRESPONDING Z VALUES FOR THE THREE MOST
COMMON CHOICES FOR C?

90%
z=

95%
z=

99%
z=
HOW DO WE CALCULATE A CONFIDENE INTERVAL?

For a sufficiently large sample size n (CLT can apply so x ̄ follows a normal
distribution) or a population that is already normally distributed, the general formula
for a level C confidence interval for the population mean μ when σ is known is given
by:
WHAT DOES THIS HAVE TO DO WITH SAMPLING DISTRIBUTIONS?

Sample means vary in value and form a sampling distribution in which not all
samples result in x -values equal to the population mean μ. We should not expect to
obtain a sample mean x (based on a specific sample) that is exactly equal to the
population mean μ.

However, we can expect the point estimate to be fairly close in value to the
population mean for a sufficiently large sample size (sampling distribution becomes
approximately normal for large sample size). (CENTRAL LIMIT THEOREM)
Recall 68-95-99.7 rule: 95% of all observations from a normal distribution will fall within ± 2
standard deviation.

If the sample size n is large enough, the sampling distribution of the sample means is
approximately normal. Our point estimate x ̄will hardly be equal to the population
mean μ, but most likely (≈ 95% of all times) fall within 2 standard deviations about
the population mean μ.
HOW DO WE INTERPRIT CONFIDENCE INTERVALS?

1- We can be C% confident that the falls in the constructed level C confidence
interval, i.e. between the lower and upper CI bound for a specific calculated example.

2- If we would take repeated samples, approximately C% of all samples taken will
include the in the long run.

3- The interpretation of a CI is always in terms of the unknown population mean μ
and never in terms of the sample mean x ̄. The sample mean x ̄, the center of every CI,
will always be included in the CI by default.
WHAT DO WE NEED TO BE CAUTIOUS OF?

Before we take a sample from a population we can say there is a C% chance, (e.g.
95% chance), that our confidence interval will include the population parameter μ if
we plan on constructing C% confidence intervals, (e.g. 95% CIs).

Once we have taken the sample, this decision is made. Our interval either does
contain μ or it does not. We just don’t know it. There is not a C% chance anymore,
all we can say is that we are C% confident, (e.g. 95% confident)