Hypothesis testing

Hypothesis Testing
The Basics

Advanced Statistics
SRSTHS
Mrs. Ma. Cristina C. Pegollo

What is Hypothesis?
 in statistics, is a claim or statement about
a property of a population
 an educated guess about the population
parameter

What is hypothesis testing?
This is the process of making
an inference or generalization on
population parameters based on
the results of the study on
samples.

What is statistical hypothesis?

It is a guess or prediction
made by the researcher
regarding the possible outcome
of the study.

Fundamentals of
Hypothesis Testing

Central Limit Theorem
If n (the sample size) is large, the
theoretical sampling distribution of
the mean can be approximated
closely with a normal distribution.

If researchers increase the samples to a considerable
number, the shape of the distribution approximates a
normal curve.

Central Limit Theorem
The Expected Distribution of Sample Means
Assuming that  = 98.6

Likely sample means

µx = 98.6

Figure 7-1 Central Limit Theorem
The Expected Distribution of Sample Means
Assuming that  = 98.6

Likely sample means

µx = 98.6

z = - 1.96 z= 1.96
or or
x = 98.48 x = 98.72

Components of a
Formal Hypothesis
Test

Null Hypothesis: H0
 This is the statement that is under
investigation or being tested.

 It is always hoped to be rejected.

 Usually the null hypothesis represents a
statement of “no effect”, “no difference”, or
, put another way, “things haven’t changed.”
 Must contain condition of equality
 =, , or 
 Reject H0 or fail to reject H0

Alternative Hypothesis: H1
This is the statement you will adopt in
the situation in which the evidence
(data) is so strong that you reject H0
 „opposite‟ of H0

 , <, >
 generally represents the idea which
the researcher wants to prove.

Example1: Null and Alternate Hypothesis


Note about Forming Your Own Claims
(Hypotheses)

If you are conducting a study and
want to use a hypothesis test to
support your claim, the claim
must be worded so that it
becomes the alternative
hypothesis.

Note about Testing the Validity of
Someone Else’s Claim

Someone else’s claim may
become the null hypothesis
(because it contains equality), and
it sometimes becomes the
alternative hypothesis (because it
does not contain equality).

Exercises:
Formulate the null and alternative hypotheses of the
following research problems
1. A manager wants to know if the average length of
time for board meetings is 3 hours.
2. The researchers want to know if the proportion of
car accidents in Balete Drive has increased from 5%
of the total car accidents recorded in Quezon City
in a day.
3. A teacher wants to know if there is a significant
difference in the academic performance between
Pasteur and Linnaeus students.
4. The registrar wants to know if the average encoding
time is lower than 30 minutes.
5. The Discipline officer wants to know if the new
policy on smoking has reduced the number of
smokers this year than the previous year.

Test Statistic
a value computed from the sample data that
is used in making the decision about the
rejection of the null hypothesis

Test Statistic
a value computed from the sample data that is used in making
the decision about the rejection of the null hypothesis
For large samples, testing claims about population means

x - µx
z=

n

Critical Region
Set of all values of the test statistic that would cause a
rejection of the
null hypothesis

Critical Region
rejection of the
null hypothesis

Critical
Region

Critical Region
rejection of the
null hypothesis

Critical
Regions

Significance Level
 denoted by 
 the probability that the
test statistic will fall in the
critical region when the null
hypothesis is actually true.
common choices are
0.05, 0.01, and 0.10

Critical Value
Value or values that separate the critical region
(where we reject the null hypothesis) from the
values of the test statistics that do not lead
to a rejection of the null hypothesis

Critical Value

Critical Value
( z score )

Critical Value

Reject H0 Fail to reject H0

Critical Value
( z score )

Two-tailed,Right-tailed,
Left-tailed Tests

The tails in a distribution are the extreme regions bounded
by critical values.

Two-tailed Test
H0: µ = 100
H1: µ  100

Two-tailed Test
H0: µ = 100
 is divided equally between
H1: µ  100 the two tails of the critical
region

Two-tailed Test
H0: µ = 100
region

Means less than or greater than

Two-tailed Test
H0: µ = 100
region

Means less than or greater than

Reject H0 Fail to reject H0 Reject H0

100

Values that differ significantly from 100

Right-tailed Test
H0: µ  100
H1: µ > 100

Right-tailed Test
H0: µ  100
H1: µ > 100

Points Right

Right-tailed Test
H0: µ  100
H1: µ > 100

Points Right

Fail to reject H0 Reject H0

Values that
differ significantly
100
from 100

Left-tailed Test
H0: µ  100
H1: µ < 100

Left-tailed Test
H0: µ  100
H1: µ < 100
Points Left

Left-tailed Test
H0: µ  100
H1: µ < 100
Points Left

Reject H0 Fail to reject H0

Values that
differ significantly 100
from 100

Conclusions
in Hypothesis Testing
 always test the null hypothesis

1. Reject the H0

2. Fail to reject the H0

 need to formulate correct wording of final conclusion

See Figure 7-4

FIGURE 7-4 Wording of Final Conclusion
Start

Does the “There is sufficient (This is the
original claim contain Yes Do Yes
you reject evidence to warrant only case in
the condition of (Original claim H0?. (Reject H0) rejection of the claim which the
equality that. . . (original claim).” original claim
contains equality
and becomes H0)
No is rejected).
(Fail to
“There is not sufficient
No reject H0)
evidence to warrant
(Original claim
rejection of the claim
does not contain
that. . . (original claim).”
equality and
becomes H1)
(This is the
Do Yes “The sample data only case in
you reject supports the claim that which the
H0? (Reject H0) . . . (original claim).” original claim
No is supported).
(Fail to
reject H0) “There is not sufficient
evidence to support
the claim
that. . . (original claim).”

Accept versus Fail to Reject
 some texts use “accept the null hypothesis

 we are not proving the null hypothesis

 sample evidence is not strong enough to warrant
rejection (such as not enough evidence to convict a
suspect)

 If you reject Ho, iit means it is wrong!

 Ifyou fail to reject Ho , it doesn’t mean it is correct –
you simply do not have enough evidence to reject it!

Type I Error
 The
mistake of rejecting the null hypothesis
when it is true.

(alpha) is used to represent the probability
of a type I error

 Example:Rejecting a claim that the mean
body temperature is 98.6 degrees when the
mean really does equal 98.6

Type II Error
 the
mistake of failing to reject the null
hypothesis when it is false.

ß (beta) is used to represent the probability of
a type II error

 Example: Failing to reject the claim that the
mean body temperature is 98.6 degrees when
the mean is really different from 98.6

Table 7-2 Type I and Type II Errors
True State of Nature
The null The null
hypothesis is hypothesis is
true false

Type I error
We decide to Correct
(rejecting a true
reject the decision
null hypothesis)
null hypothesis

Decision
Type II error
We fail to Correct (rejecting a false
reject the decision null hypothesis)
null hypothesis


Controlling Type I and Type II Errors
 Forany fixed , an increase in the
sample size n will cause a decrease in 

 For any fixed sample size n , a decrease
in  will cause an increase in .
Conversely, an increase in  will cause a
decrease in  .

 Todecrease both  and , increase the
sample size.

Approaches in Hypothesis Testing

1. Critical Value Approach

2. P-value Approach

Example:
The average score in the final examination
in College Algebra at ABC University is
known to be 80 with a standard deviation
of 10. A random sample of 39 students was
taken from this year’s batch and it was
found that they have a mean score of 84.
Test at 0.05 level of significance.

Hypothesis testing

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