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Inference for
the Mean of a
Population
Section 11.1
Conditions for
Inference about a Mean
 SRS
of size n from the population of interest
 Observations
 Observations
must be independent
must have a normal distribution
with mean µ and standard deviation σ
What if σ isn’t given?
 Because
σ is usually unknown, we
estimate it by the sample deviation s.
Standard Error
 The
standard error of the sample mean 𝑥 is
𝑠
.
𝑛
 When
we know the value σ, we base
confidence intervals and tests for µ on the
z - statistic:
𝑧=
𝑥−𝜇
𝜎/ 𝑛
The z – statistic has a normal distribution.
 When
we do not know σ, we substitute
the standard error 𝑠/ 𝑛. This statistic does
not have a normal distribution…
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
𝑥−𝜇
𝑡=
𝑠/ 𝑛
has the t distribution with n – 1 degrees of
freedom.
Different t distributions?
 There
is a different t distribution for each
sample size.
So how do we determine
which one we use?
 Degrees
of freedom
n -1 , where n is the sample size.
Example 11.1, p. 619
Using the “t Table”
What critical value t* from Table C (t Table) would you use for
a t distribution with 18 degrees of freedom having probability
0.90 to the left of t*?
Example 11.1, p. 619
Using the “t Table”
What critical value t* from Table C (t Table) would you use for
a t distribution with 18 degrees of freedom having probability
0.90 to the left of t*?
Before we start, we will need the tail probability.
.90 probability
Tail area = .10
Df = 18, Tail Area = .10
Example 11.1, p. 619
Using the “t Table”
What critical value t* from Table C (t Table) would you use for
a t distribution with 18 degrees of freedom having probability
0.90 to the left of t*?
So the desired critical value is t* = 1.330.
Example 11.1, p. 619
Using the “t Table”
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 of t* should you use?
Df = 12 – 1 = 11
Tail Area?
We don’t need it…look at the bottom of the chart.
Example 11.1, p. 619
Using the “t Table”
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 of t* should you use?
Example 11.1, p. 619
Using the “t Table”
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 of t* should you use?
So the desired critical value is t* = 2.201.
Homework:
 P.
619: 11.1 a, 11.2, 11.4
Due: Tuesday
FYI: I have a meeting Friday morning, 3/8. I
will have to cancel tutoring that day.