The document provides an introduction to hypothesis testing, including its real-life applications, key definitions, and structure. It defines hypothesis testing as the process of testing the validity of a statistical hypothesis based on a random sample from a population. The document outlines the common steps in hypothesis testing: 1) stating the null and alternative hypotheses, 2) choosing a significance level, 3) determining the test statistic and decision criteria, 4) rejecting or failing to reject the null hypothesis, and 5) drawing a conclusion. It also defines important terminology like population mean, null and alternative hypotheses, test statistic, significance level, critical region, and p-value. Real-life examples from pharmaceutical testing and legal cases are provided to illustrate the motivation for hypothesis

1. I N T R O D U C T I O N
- R E A L L I F E A P P L I C A T I O N S
- D E F I N I T I O N S
- S T R U C T U R E
Hypothesis Testing

2. Lesson Objectives
By the end of this lesson, you should be able to:
1. Describe real-life examples to explain the motivation behind
hypothesis testing.
2. State the definitions of the terminologies.
3. Identify the steps in the structure of hypothesis testing.
4. Explain, in the context of a given scenario, the meaning of the
terminologies.
5. Formulate the null (H0) and alternative (H1) hypotheses of a given
scenario.

3. What is hypothesis testing?
 A statistical hypothesis is an assumption about the value
of a population parameter.
 Hypothesis testing is…
the process of testing the validity of the statistical hypothesis
based on a random sample drawn from the population.

4. What is hypothesis testing?
Data from
the sample is
collected.
REJECT
or
DO NOT REJECT
Modified from http://www2.seminolestate.edu/jmclaughlin/zSTAT/lecture15.htm
Does a sample mean of 327 ml
provide sufficient evidence to
justify that mean fill of the
population is 330 ml?
A plant manager
believes that:
mean fill of the
population of Coca
Cola cans is 330ml
327

5. Why study hypothesis testing?
 Hypothesis testing is performed regularly in many industries.
 Pharmaceutical industry
must perform many hypothesis tests on new drug products before they
are deemed to be safe and effective by the federal Food and Drug
Administration (FDA). In these instances, the drug is hypothesized to be
both unsafe and ineffective. Then, if the sample results from the studies
performed provide “significant” evidence to the contrary, the FDA will
allow the company to market the drug.
 Basis of the legal system
in which judges and juries hear evidence in court cases. In a criminal
case, the hypothesis in the American legal system is that the defendant is
innocent. Based upon the totality of the evidence presented in the trial, if
the jury concludes that “beyond a reasonable doubt” the defendant
committed the crime, the hypothesis of innocence will be rejected and
the defendant will be found guilty. If the evidence is not strong enough,
the defendant will be judged not guilty. (Innocent until proven guilty!)
www.prenhall.com/divisions/bp/app/chapters/groebner7/Chapter09.pdf

6. Why study hypothesis testing?
 Business Statistics
Suppose a Coca-Cola plant produces approximately
400,000 cans of Coke products daily.
Each can is supposed to contain 330ml of fluid.
However, like all processes, the automated filling machine is subject to variation and
each can will contain either slightly more or less than the 330ml target. The important
thing is that the mean fill is 330ml.
The plant quality manager‟s initial belief is that the filling process is providing an
average fill of 330ml. This is his hypothesis.
Every two hours, the plant quality manager selects a random sample of cans and
computes the sample mean. If the sample mean is “close” to 330ml, then the
sample data would tend to support the hypothesis and the machine would be allowed
to continue filling cans. However, if the sample mean is “significantly” higher or lower
than 330ml, the data would refute the hypothesis and the machine would be stopped
and adjusted.

7. Definitions
 Population mean, µ
 Null Hypothesis, H0
This is a statement about a parameter of the population which is initially, or
conventionally believed to be true.
It is denoted by H0.
In our syllabus, H0 will always be stated as an equality claim, for example H0 : µ = 25
 Alternative Hypothesis , H1
It is any other hypothesis when the null hypothesis is rejected.
It is denoted by H1.
Examples of the 3 possible H1 in our syllabus are µ > 25 , µ < 25 and µ ≠ 25 .

8. Definitions
 Test Statistic
It is a sample statistic which is used to make the decision whether or not to reject
the null hypothesis.
A random sample is selected from the given population and a relevant test statistic with
respect to the parameter of interest is observed.
 Level of Significance
It is the largest probability of wrongly rejecting H0 (when H0 is in fact true).
In other words, it refers to how much error we allow to wrongly reject H0.
If the level of significance is 5%,
then there is a probability of 0.05 that we reject H0 when H0 is in fact true.
(We are 95% confident that we would make the right decision.)
Important
definition!

9. Definitions
Decision Criteria (we will learn more about this in the next lesson)
 Critical Region
It is the set of values of the test statistic for which H0 will be rejected.
This set of values is determined by the level of significance.
The boundaries of the critical region are called the critical values.
 p-value
It is the smallest level of significance for the null hypothesis to be rejected.
It represents the probability of obtaining a test statistic as extreme as the one calculated
from the sample, assuming that the null hypothesis is true.
In other words, if we set our level of significance at this p-value or higher, we will end up
rejecting the null hypothesis.
Important
definition!

10. Structure
Suppose a Coca-Cola plant produces
approximately 400,000 cans of Coke
products daily. Each can is supposed to
contain 330ml of fluid.
Step 0
Define the parameter to be tested
Let µ be the mean fill of the population of
Coca-Cola cans
Step 1
Decide the null and alternative
hypotheses
H0: µ= 330
H1: µ≠ 330
Step 2
State the level of significance
Level of significance: 5%
However, like all processes, the automated filling
machine is subject to variation and each can will
contain either slightly more or less than the 330ml
target. The important thing is that the mean fill is
330ml.
The plant quality manager‟s initial belief is that the
filling process will provide a mean fill of 330ml. This
is his null hypothesis.
He wishes to check if the mean fill differs from
330ml. This is his alternative hypothesis.
The plant manager decides his level of tolerance for
error:
He will allow at most a 5% chance of wrongly
rejecting H0 (i.e. mean fill is 330) when H0 is true
(i.e. mean fill is indeed 330).

11. Structure
The mean fill of Coca Cola cans is normally
distributed with mean 330 and variance 2.6.
Step 3
State the distribution of the
population mean
[Apply Sampling Theory]
Let X be the fill of a Coca Cola can.
Under H0,
Step 4
State the test statistic
[Standardize the distribution of ]
2.6
~ N 330,
50
X
2.
330
,
6 /
N(0,1)
50/
Z
X X
Z
n

X
The plant quality manager selects a
random sample of 50 cans and
computes the sample mean, 327x

12. Structure
Step 5
Decision Criteria
Calculate test statistic and p-value
[using the sample mean ]
More on this the next lesson
Step 6
Reject H0/Do not reject H0
Step 7
Conclude
327x
Based on the decision criteria,
there are two possible conclusions:
• Possibility #1
(the sample mean is “significantly” higher or
lower than 330ml)
Reject H0.
There is sufficient evidence at 5% level
of significance to conclude [H1 ] the mean
fill of Coke cans is not equal to 330ml.
• Possibility #2
(the sample mean is NOT “significantly”
higher or lower than 330ml)
Do not reject H0.
There is insufficient evidence at 5%
level of significance to conclude [H1 ] the
mean fill of Coke cans is not equal to 330ml.
Compare test
statistic with
critical value
Compare p-
value with level
of significance

13. Types of tests
1. One tailed test
A one-tailed test looks for a definite increase (upper-tail) or
a definite decrease (lower-tail) in the parameter used.
2. Two tailed test
This test looks for any difference or change in the parameter used.
The type of test we choose, together with the level of significance, determines the critical
region of the hypothesis test.
Type of test
One-tailed test
Two-tailed test
Upper-tail Lower-tail
Example
H0: µ = 330
H1: µ > 330
H0: µ = 330
H1: µ < 330
H0: µ = 330
H1: µ ≠ 330

14. Formulating H0 and H1
In the following examples, formulate the null and alternative hypotheses.
 A random sample of JC students was each given an IQ test to decide whether
JC students have a mean IQ of 100 or greater than 100.
 Suppose that, according to national standards, 5 year old children of height
87.5 – 92.5 cm have a mean weight of 13 kg. The weights of a random sample of
5 year old children of height 87.5 – 92.5 cm are collected. Are 5 year old
children of height 87.5 – 92.5 cm below average weight for their height?
1
0H : 100
H : 100
1
0H : 13
H : 13
• Recall that the null hypothesis (H0) is stated as an equality
claim. Assume that mean IQ of JC students is 100.
• Want to prove: mean IQ of JC students is greater than 100.
• Recall that the null hypothesis (H0) is stated as an equality
claim. Assume that average weight of 5 year olds of height 87.5
– 92.5 cm is 13 kg.
• Want to prove: average weight of 5 year olds of height 87.5 –
92.5 cm is less than 13 kg.

15. Formulating H0 and H1
 A machine produces components of mean length 10 cm if set up correctly. A
random sample of components produced is taken, and the length of each
component is measured. Is the machine set up correctly?
 The principal of a school claims that the mean „A‟ Level score of all its students
is at least 80. Has the principal overstated the mean?
1
0H : 10
H : 10
1
0H : 80
H : 80
• Recall that the null hypothesis (H0) is stated as an equality
claim. Assume machine is set up correctly: mean length 10 cm.
• We want to know if the machine is set up incorrectly (we are
only interested in whether it differs from 10 cm, not interested
in whether it is „more‟ or „less‟ than 10 cm.)
Want to prove: mean length of components is not 10 cm
• Recall that the null hypothesis (H0) is stated as an equality
claim. Assume mean „A‟ Level score of all students is 80.
• In order to justify that the principal has indeed overstated the
mean score, we need to prove that the mean „A‟ Level score
should be less than 80.

16. Task
 Compare hypothesis testing at „5% level of significance‟ and
„10% level of significance‟, which is a stricter test? Explain
why.
 Is 'do not reject the null hypothesis' equivalent to 'accept
the null hypothesis'?
Check if you have met the lesson objectives!