Lecture 5.pptx - Introduction to Statistics Hypothesis...

Introduction to Statistics

Hypothesis Testing

Hypothesis Testing
•
We are often interested in testing the validity of some
statement
•
Suppose you want to determine whether the mean level of
a driver’s blood alcohol exceeds the legal limit after two
drinks….? Is the underlying mean return on this mutual
fund different from the underlying mean return on its
benchmark?
•
You are interested in making an inference about how the
value of a parameter relates to a specific numerical value

Hypothesis Testing
•
Hypothesis testing: part of the branch of statistics known as
statistical inference
•
A hypothesis is a statement about one or more populations
Hypothesis testing allows us to use a sample to decide between
two statements made about a population characteristic.
These two statements are called the Null Hypothesis and the
Alternative Hypothesis .

Terms Introduce in Prior Chapter
•
Population  all possible values
•
Sample  a portion of the population
•
Statistical inference  generalizing from a sample to
a population with calculated degree of certainty
•
Two forms of statistical inference
•
Hypothesis testing
•
Estimation
•
Parameter  a characteristic of population, e.g., population
mean µ
•
Statistic  calculated from data in the sample, e.g., sample
mean ( )
x

Review: Distinctions Between
Parameters and Statistics
Parameters Statistics
Source Population Sample
Notation Greek (e.g., μ) Roman (e.g., )
Vary No Yes
Calculated No Yes
Parameters Statistics
Source Population Sample
Notation Greek (e.g., μ)
Vary No Yes
Calculated No Yes

The Elements of a Test of
Hypothesis
The null hypothesis
H
0
: The Null Hypothesis
This is the hypothesis being tested. It is the statement
that we are attempting to disprove.
“Portfolio returns are 5%”
“My portfolio is no riskier than the market”

The Elements of a Test of
Hypothesis
The Alternative Hypothesis
H
A
: The Alternative Hypothesis
This is the hypothesis or claim which we initially assume to
be false but which we may decide not to reject if there is
sufficient evidence .
If we disprove H
0,
we can accept the alternative hypothesis.

The Legal Analogy
•
This procedure is familiar to us already from the legal
system “Innocent until proven guilty”.
•
The null hypothesis (not guilty) is only rejected in
favour of the alternative (guilty) if there is sufficient
evidence of this “beyond reasonable doubt”
•
The null hypothesis will not be rejected unless the data
provide convincing evidence that it is false.
•
The alternative hypothesis is that which will be
accepted only if the data provide convincing
evidence of its truth

The Hypotheses
• Null Hypothesis H
0
: ‘There is no difference
between the average number of streams per square
kilometer and the bedrock type .‘ This is the
hypothesis to be tested
•
Alternative Hypothesis H
A
or H
1
: ‘There is a
difference between the average number of
streams per square kilometer and the bedrock
type .'

Steps in Hypothesis Testing
1. Stating the hypotheses
2. Identify the appropriate test statistic and it’s
probability distribution
3. Specifying the significance level
4. Stating the decision rule
5. Collecting the data and calculating the test statistic
6. Making the statistical decision

Formulating the Hypotheses