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CHAPTER 11
Section 11.1 – Inference for the Mean of a
Population
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 one-sample z statistics
𝑥−𝜇
𝑧= 𝜎
𝑛


When 𝑠we do not know σ, we substitute 𝜎the standard
error 𝑛 of 𝑥 for its standard deviation 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,
single-peaked, 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 PROCEDURES
 Draw
an SRS of size n from a population
having unknown mean μ. A level C confidence
interval for μ is
𝑠
𝑥 ± 𝑡∗
𝑛
 Where
𝑡 ∗ is the upper (1 - C)/2 critical value for
the t(n – 1) distribution. This interval is exact
when the population distribution is normal and
is approximately correct for large n in other
cases.
 The
test the hypothesis H0 : μ = μ0 based on an
SRS of size n, computed the one-sample t
statistic
𝑥−𝜇
𝑡= 𝑠0
𝑛
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 - 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?


𝑡 ∗ = 1.330
Now suppose you want to construct a 95%
confidence interval for the mean 𝜇 of a population
based on an SRS of size n = 12. What critical value
𝑡 ∗ should you use?

𝑡 ∗ = 2.201
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 onesample t statistic has the t distribution
with n – 1 degrees of freedom.
𝑥−𝜇
𝑡= 𝑠
𝑛
THE ONE-SAMPLE T PROCEDURE (CONTINUED…)
 In
terms of a variable T having then
t(n – 1) distribution, the P-value for a
test of Ho against
Ha: μ > μo is P( T ≥ t)
Ha: μ < μo is P( T ≤ t)
Ha: μ ≠ μo is 2P( T ≥ |t|)
 These
P-values are exact if the population
distribution is normal and are approximately
correct for large n in other cases.
EXAMPLE 11.2 - AUTO POLLUTION

See example 11.2 on p.622
Minitab stemplot of the
data (page 623)
The one-sample t confidence interval has the form:
estimate ± 𝑡 ∗ SEestimate
(where SE stands for “standard error”)

Homework: P.619 #’s 1-4, 8 & 9