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# Test of a Statistical Hypothesis

Hypothesis

## Hypothesis testing begins with an assumption, called

Hypothesis, that we make about a population parameter.
Then we collect sample data, produce sample statistics,
and use this information to decide how likely it is that our
hypothesized population parameter is correct.
Suppose a manger of a large shopping mall tells us that
the average work efficiency of her employees is at least 90
percent.
How can we test the validity of her Hypothesis?
If we drop the price of this car model by \$1,500, we will
sell 50,00 cars this year, is a hypothesis.
Hypothesis testing is about making inferences about a
population from only a small sample.

Testing of Hypothesis
In hypothesis testing, we must state the assumed or
hypothesized value of the population parameter before we
begin sampling
The assumption we wish to test is called the null hypothesis
and is symbolized H0

## Example: Suppose we want to test the hypothesis that population mean is

equal to 500.
H0 =500
The null hypothesis is that the population mean is equal to 500.
Why it is called Null Hypothesis?
The term null hypothesis arises from earlier agricultural and
medical applications of statistics. In order to test effectiveness
of a new fertiliser or a new drug, the tested hypothesis ( the
null hypothesis ) was that it had no effect, that is, there was no
difference between treated and untreated samples.

Alternative Hypothesis
If our sample results fail to support the null hypothesis, we
must conclude that something else is true.
Whenever we reject the hypothesis, the conclusion we do
accept is called the alternative hypothesis and is Symbolized H 1

Example: For H0 =500 ( the null Hypothesis is that the population mean is
equal to 500)
We will consider three possible alternative hypothesis
H1 500 ( The alternative hypothesis is that the population mean is not equal
to 500)
H1 > 500 (The alternative hypothesis is that the population mean is greater
than 500
H1 <500 (The alternative hypothesis is that the population mean is less than
500

Significance Level
The purpose of hypothesis testing is not to question the computed value
of the sample statistics but to make a judgment about the difference
between the sample statistics and a hypothesized population parameter.
The next step after stating the null and alternative hypothesis, then, is to
decide what criterion to use for deciding whether to accept or reject the
null hypothesis

## If we assume the hypothesis is correct, then the significance level will

indicate the percentage of sample means that is outside certain limits.

## How to interpret a 5 percent level of significance?

It means in 95 percent of the area, there is no significant difference
between the observed value of the sample statistics and the hypothesized
value of the population parameter.
In the remaining 5 percent, a significant differences does exist.
Accepting a null hypothesis do not prove that our Null Hypothesis is
true, it simply does not provide statistical evidence to reject it. Why
Because the only way in which the hypothesis can be accepted with
certainty is for us to know the population parameter.

It means simply that when sample data do not cause us to reject a null
hypothesis, we behave as if that hypothesis is true.

Selecting
Significance
Level/Confidence
Limits

## The limits (or range) within which the hypothesis

should lie with specified probabilities are called the
confidence limits.
. If sample values lie between the confidence
limits, the hypothesis is accepted. If it does not,
the hypothesis is rejected at the specified level of
significance.

Type I Error
Rejecting a null Hypothesis when it is true is
called a Type I error
Type I error is denoted by a (alpha) where
a = Probability of type I error
= Probability (rejecting Ho | Ho is true ).

## After deciding what level of significance to

use, or next task in hypothesis testing is to
determine the appropriate probability distribution

Conditions for Using the Normal and t Distributions in Testing Hypotheses about
means
When the population

is

## Conditions for Using the Normal and t Distributions in Testing Hypotheses

Standard Deviation is Standard Deviation
about means
When the
When thenot
population
Know
Know
population
Standard Deviation is
Standard
not Know
Deviation is
Know

## size n is larger than 30 Normal

Normal Distribution
Sample size n Sample
is larger
than 30
Normal
Distribution,
Normal Distribution
Distribution, Z
Z table
table
Sample size n is 30 or less and Normal
t-distribution
assume the population is
Distribution,
t-table
Z tableDistribution,
n normal
is 30 ororapproximately
less and so Normal
t-distribution

Sample size
assume the population is normal Z table
or approximately so

t-table

## Two-Tailed and One-Tailed Tests of

Hypotheses

A
two-tailed test of a hypothesis will reject the null
hypothesis if the sample mean is significantly higher than or
lower than the hypothesized populations mean
Thus, in a two-tailed test, there are two rejection regions.
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