Outlines
Statistical Tests (t Test for a Mean)
Mike Renfro
December 9, 2004
Mike Renfro
Statistical Tests (t Test for a Mean)
Outlines
Review of Previous Lecture
Statistical Tests (t Test for a Mean)
Statistical Tests (Hypothesis Testing)
Mike Renfro
Statistical Tests (t Test for a Mean)
Outlines
Review of Previous Lecture
Statistical Tests (t Test for a Mean)
Statistical Tests (t Test for a Mean)
Introduction
The t Test for a Mean
Examples
Much of this material is from Allan Bluman’s Elementary
Statistics: A Brief Version, Second Edition.
Mike Renfro
Statistical Tests (t Test for a Mean)
Review of Previous Lecture
Introduction
Steps for Hypothesis Testing
Define a population
State hypotheses and significance level
Perform calculations, reach a conclusion.
Statistical Tests (Normal Distribution)
Introduction
Examples
Mike Renfro
Statistical Tests (t Test for a Mean)
Introduction
The t Test for a Mean
Part II
Statistical Tests (t Test for a Mean)
Mike Renfro
Statistical Tests (t Test for a Mean)
Introduction
The t Test for a Mean
Why Another Statistical Test?
The normal distribution and the z test are most suitable when:
The sample size is over 30, or
The sample size is under 30 and the population standard
deviation is known beforehand.
Under other conditions, such as when the sample size is under
30 and the population standard deviation is not known
beforehand, another technique must be used.
Mike Renfro
Statistical Tests (t Test for a Mean)
Introduction
The t Test for a Mean
Examples
The t Test for a Mean
The t test is a statistical test for the mean of a population. It
should be used whenever all of the following criteria apply:
the sample size n < 30,
the population standard deviation σ is unknown, and
the population is known to be normally distributed
The formula for the t test is
t=
X̄ − µ
√
s/ n
There is also a new concept for the t test known as degrees of
freedom. The probability distribution function for the t distribution
isn’t a single curve, but a family of them. We use the value of the
degrees of freedom to select the right distribution curve. Normally,
the value for degrees of freedom is n − 1.
Mike Renfro
Statistical Tests (t Test for a Mean)
Introduction
The t Test for a Mean
Examples
The t Test for a Mean
If this t value falls into the critical region, we reject the null
hypothesis, and conclude that there is enough evidence to
support the idea that there is a significant difference between
the sample and the overall population.
Rao gives various values for critical values of t for different
significance levels α and different degrees of freedom in Table
G.2.
All of the α values given in Table G.2 are for one-tailed tests.
If you’re conducting a two-tailed test, double Rao’s α values
before doing your table lookup.
Also, Rao refers to the degrees of freedom as n in Table G.2.
This may be confusing, since n is traditionally used as a
sample size.
Mike Renfro
Statistical Tests (t Test for a Mean)
Introduction
The t Test for a Mean
Examples
Example 1
Back to the medical researcher. The mean pulse rate of the entire
population of adult men is 70 beats per minute (bpm). A sample
of 10 adult male patients is given a new drug, and after several
minutes, their mean pulse rate is measured at 75 bpm with a
standard deviation of 8 bpm. With a level of significance of
α = 0.01, is this change in pulse rate due to random chance?
Mike Renfro
Statistical Tests (t Test for a Mean)
Introduction
The t Test for a Mean
Examples
Example 1 Solution (1)
State the null hypothesis: “There is no significant difference
in pulse between these patients and the population as a
whole.” (H0 : µ = 70)
State the alternative hypothesis: “There is a significant
difference in pulse between these patients and the population
as a whole.” (H1 : µ 6= 70).
Our degrees of freedom is equal to n − 1, so d.f. = 9.
The null hypothesis is given in the form corresponding to a
two-tailed test (H0 : µ = k, H1 : µ 6= k), and for α = 0.01 and
d.f. = 9, the critical t values are t = ±3.250.
Mike Renfro
Statistical Tests (t Test for a Mean)
Introduction
The t Test for a Mean
Examples
Example 1 Solution (2)
Calculate the test t value:
t=
X̄ − µ
√
s/ n
with X̄ = 75, µ = 70, s = 8, n = 10.
In this case, t = 1.98. Since t falls within the noncritical
region, our conclusion is to not reject the null hypothesis.
We conclude that there is not sufficient evidence to indicate
that the change in pulse rate is due to anything other than
random chance.
Mike Renfro
Statistical Tests (t Test for a Mean)