Download Section 11.1 Part 1 – Inference for the Mean of a Population

Section 11.1 Part 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 𝑥̅ .

The sampling distribution of 𝑥̅ has μ as its mean.


That is an __________________________________ of the unknown μ.
In the previous chapter we make the unrealistic assumption that we knew the value of σ. In practice, _____
___________________.
Conditions for Inference About a Mean

Our data are a ______________________________________________ from the population of interest.
This condition is very important.

Observations from the population have a______________________________________________________.
In practice, it is enough that the distribution be ___________________________________________ unless
the ________________________________.

Both μ and σ are unknown parameters.
Standard Error

When the standard deviation of a statistic is ____________________________________, the result is
called the ________________________________of the statistic.

The standard error of the sample mean 𝑥̅ is _________________________
The t distributions

When we know the value of σ, we base confidence intervals and tests for μ on _______________________
𝑧=

𝑥̅ − 𝜇
𝜎
√𝑛
When we do not know σ, we substitute the ____________________ of 𝑥̅ for its standard deviation _______

The statistic that results does not have a normal distribution. It has a distribution that is new to us, called a
___________________________.

The density curves of the t distributions are _________________________________to the standard normal
curve. They are_______________________________________________________________________

The spread of the t distribution is a bit _______________________ that of the standard normal
distribution. The t have ___________________________in the ___________________and ___________ in
the __________________________ than does the standard normal.

As the _______________________________________________________, the t(k) density curve
approaches the ____________________ 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. ________________________________________has the t distribution with _______________
_______________________________.
𝑡=
𝑥̅ − 𝜇
𝑠
√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 ____________ 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 Procedures

Draw an SRS of size n from a population having unknown mean μ. A level C confidence interval for μ is
𝑥̅ ± 𝑡 ∗
𝑠
√𝑛

Where 𝑡 ∗ is the upper _____________ critical value for the _________________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 _________________________.

In terms of a variable T having then t(n – 1) distribution, the P-value for a test of Ho against

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

The one-sample t confidence interval has the form:
estimate ± 𝑡 ∗ SEestimate
(where SE stands for “standard error”)