Download `null` hypothesis

Chapter 3
Hypothesis
Testing
Curriculum Object
• Specified the problem based the form of hypothesis
• Student can arrange for hypothesis step
• Analyze a problem bassed for hypothesis
Introduction…
• In addition to estimation, hypothesis testing is a
procedure for making inferences about a
population.
Population
Sample
Inference
Statistic
Parameter
11.3
What is Hypothesis Testing?
• Sample information can be used to obtain point
estimates or confidence intervals about population
parameters
• Alternatively, sample information can be used to test
the validity of conjectures about these parameters
RESume
• A hypothesis is a statement about a population
parameter from one or more populations
• Statistically testable hypotheses are formulated based
on theories that are used to make predictions
• A hypothesis test is a procedure that
– States the hypothesis to be tested
– Uses sample information and formulates a
decision rule
– Based on the outcome of the decision rule the
hypothesis is statistically validated or rejected
Concepts of Hypothesis Testing
• There are two hypotheses.
pronounced
H “nought”
•
•
H0: — the ‘null’ hypothesis
H1: — the ‘alternative’ or ‘research’ hypothesis
• The null hypothesis (H0) will always state that the
parameter equals the value specified in the alternative
hypothesis (H1)
11.6
Null hypothesis (H0)
– The hypothesis that there were no effects is called
the NULL HYPOTHESIS.
– The null hypothesis states that in the general
population there is no change, no difference, or
no relationship.
– In the context of an experiment, H0 predicts that
the independent variable (treatment) will have
no effect on the dependent variable for the
population.
– Form :
H0: μA- μB=0 or μA= μB
Alternative hypothesis (H1)
– The alternative hypothesis (H1) states that there is
a change, a difference, or a relationship for the
general population.
– H1 is a statement of what a statistical hypothesis
test is set up to establish.
– Form :
H1: μA≠ μB
• For example
– H1: the two drugs have different effects, on average.
– H1: the new drug is better than the current drug, on
average.
Type I Error
• A type I error is made when the researcher
rejected the null hypothesis when it should not
have been rejected.
• For example,
– H0: there is no difference between the two
drugs on average.
• A type I error would occur if we concluded that
the two drugs produced different effects when in
fact there was no difference between them.
• The hypothesis test procedure is therefore adjusted
so that there is a guaranteed 'low' probability of
rejecting the null hypothesis wrongly; this
probability is never 0.
• This probability of a type I error can be precisely
computed as
P(type I error) = significance level = α
• The exact probability of a type II error is generally
unknown.
Type II Error
• A type II error is made when the null hypothesis
is accepted when it should have been rejected.
• For example,
– H0: there is no difference between the two drugs on
average.
• A type II error would occur if it was concluded that the
two drugs produced the same effect, i.e. there is no
difference between the two drugs on average, when in
fact they produced different ones.
• The probability of a type II error is generally unknown,
but is symbolised by and written
– P(type II error) = β
RESUME
The following table gives a summary of possible results of
any hypothesis test:
– Decision
Reject H0
Don't reject H0
H0 Type I Error
Right decision
– Truth
H1 Right decision Type II Error
  P(Type I Error )   P(Type II Error )
Goal: Keep ,  reasonably small
Example
So, We have 3 kinds of hypothesis :
Steps in aplllying Hypothesis testing :
i.
State The Hypothesis
a. H 0 :    0
H1 :   0
b. H 0 :    0
H1 :   0
c. H 0 :    0
H1 :   0
ii. Choose a significance level
Two Tailed Test
• H0: μ = μo
• H1: μ ≠ μo
Rejection Region H0
Rejection Region H0
Acceptance Region H0
½α
iii. H0 accepted if: -z1/2α < z < z1/2 α
½α
One Tailed Test (right)
• H0: μ = μo
• H1: μ > μo
Critical region
Rejection Region H0
Acceptance region H0
α
iii. H0 accepted if : z ≤ z α
One tailed (left)
• H0: μ = μo
• H1: μ < μo
(critical region)
Rejection Region H0
Acceptance Region H0
α
iii. H0 accepted if: z ≥ -zα
iv. Calculation :
Z
X  0

n
X  0
Z
,  unknown
s
n
Suppose :
48.5  x  51.5
  PType I Error 
 PReject H 0 , H 0 is true 

 
 P X  48.5, with   50  P X  51.5, with   50

suppose n  10,   2.5,
48.5  50
51.5  50
 1.90
z2 
 1.90
2.5 / 10
2.5 / 10
  PZ  1.90  PZ  1.90  0.028717  0.028717  0.057434
z1 
Implies that 5.76% of all random samples
would lead to rejection of the Ho
when Ho is true


  P 48.5  X  51.5,   52
48.5  52
z1 
 4.43
2.5 / 10
51.5  52
z2 
 0.63
2.5 / 10

  P 48.5  X  51.5,   52
 P z 2   P z1 
 0.2643  0.0000
 0.2643


I. Test on the Mean of a normal Distribution, variance known
• Suppose that we wish to test the hypothesis :
• Test Statistics :
Example