AP STATISTICS
LESSON 11 – 1
(DAY 1)
INFERENCE FOR THE MEAN OF A
POPULATION
ESSENTIAL QUESTION:
What are the procedures for making
inferences when the populations
standard deviation is unknown?
Objectives:
 To become familiar with the procedures for
inference when the populations standard
deviation is unknown.
 To use t calculations and procedures that will
lead to making inferences.
Introduction
This chapter describes confidence
intervals and significance tests for the
mean of a single population and for
comparing the means of two
populations.
Inference for the Mean of a
Population

Confidence intervals and tests of significance
for the mean μ of a normal population are
based on the sample mean x. The sampling
distribution of x has μ as its mean. That is an
unbiased estimator of the unknown μ.
 In the previous chapter we make the
unrealistic assumption that we knew the
value of σ. In practice, σ is unknown.
Conditions for Inference
About a Mean

Our data are a simple random sample (SRS)
of size n from the population of interest. This
condition is very important.
 Observations from the population have a
normal distribution with mean μ and standard
deviation σ. In practice, it is enough that the
distribution be symmetric and single-peaked
unless the sample is very small.
 Both μ and σ are unknown parameters.
Standard Error
When the standard deviation of a
statistic is estimated from the data, the
result is called the Standard Error of the
statistic. The standard error of the
sample mean x is s/√ n.
The t distributions
When we know the value of σ, we base
confidence intervals and tests for μ on onesample z statistics
z=x–μ
σ/√n
When we do not know σ, we substitute the
standard error s/√ n of x for its standard
deviation σ/√ n. The statistic that results does
not have a normal distribution. It has a
distribution that is new to us, called a t
distribution.
t distributions (continued…)

The density curves of the t distributions are
similar in shape to the standard normal curve.
They are symmetric about zero, singlepeaked, and bell shaped.
 The spread of the t distribution is a bit greater
than that of the standard normal distribution.
The t have more probability in the tails and
less in the center than does the standard
normal.
 As the degrees of freedom k increase, the t(k)
density curve approached the N(0,1) curve
ever more closely.
The One-sample t Statistic
and the t Distribution
Draw an SRS of size n from a population that
has the normal distribution with mean μ and
standard deviation σ. The one-sample t
statistic
t= x–μ
S/√n
has the t distribution with n – 1 degrees of
freedom.
Degrees of Freedom

There is a different t
distribution for each sample
size. We specify a particular
t distribution by giving its
degree of freedom.

The degree of freedom for
the one-sided t statistic come
from the sample standard
deviation s in the
denominator of t.

We will write the t distribution
with k degrees of freedom as
t(k) for short.
Example 11.1 Page 619
Using the “t Table”
What critical value t* from Table C (back
cover of text book, often referred to as
the “t table”) would you use for a t
distribution with 18 degrees of freedom
having probability 0.90 to the left of t?
More Practice…
What critical value t* from Table C should be used
for a confidence interval for the mean of the
population in each of the following situations?
1. A 90% confidence interval based on n = 12
observations?
2. A 95% confidence interval froman SRS of 30
observations
3. An 80% confidence interval from a sample of size 18