GRADUATE SCHOOL

Ateneo de Naga University

Naga City

Course Code: MA216

Course Title: Inferential Statistics
Week No.5
Ms. Yumi Vivien V. De Luna

Topic: Test of Hypothesis

I. OBJECTIVES:
At the end of the lesson, the students are expected to:
a. illustrate: (i) null hypothesis, (ii) alternative hypothesis, (iii) level of significance,
(iv) rejection region, (v) types of errors in hypothesis testing; and
b. differentiate a One-tailed from Two-tailed test

II. LESSON

Definition 19: Hypothesis testing is a decision-making process for evaluating

claims about a population. In hypothesis-testing, the researcher must define the
population under study, state the particular hypotheses that will be investigated,
give the significance level, select a sample from the population, collect the data,
perform the calculations required for the statistical test, and reach a conclusion.

Hypotheses concerning parameters such as means and proportions can be

investigated. There are two specific statistical tests used for hypotheses concerning
means: z test and t test. Recall from previous discussions that the z test is used when
n ≥ 30, σ is known; when n ≥ 30, σ is unknown; or when n<30 , σ is known while the t
test is used when n<30 , σ is unknown.
The three methods used to test hypotheses are
1. The traditional method*
2. The confidence interval method
3. The p-value method

The traditional method will be used here. It has been used since the hypothesis
testing method was formulated. The second method, the confidence interval method

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illustrates the relationship between hypothesis testing and confidence intervals. A

newer method, called the p-value method, has become popular with the advent of
modern computers and high-powered statistical calculators. Since critical p values are
assigned by computers, which we will not be using in this part of the subject, we will
just settle in using the traditional method of hypothesis testing.
Every hypothesis-testing situation begins with the statement of a hypothesis.
Definition 20: A statistical hypothesis is a conjecture about a population
parameter. This conjecture may or may not be true.
There are two types of statistical hypothesis for each situation: the null and the
alternative hypothesis.
Definition 21: Null hypothesis, H o , is a statistical hypothesis that states that there
is no significant difference between a parameter and a specific value, or that there
is no significant difference between two parameters. Alternative hypothesis, H 1
or H a , is a statistical hypothesis that states the existence of a significant difference
between a parameter and a specific value, or states that there is a significant
difference between two parameters. Note: A significant difference is a difference
that cannot be explained by the sampling error.
To state hypotheses correctly, researchers must translate the conjecture or claim
from words into mathematical symbols. The basic symbols and their equivalent
phrases that are used in hypotheses and conjectures are given below.
Hypothesis-Testing Common Phrases
>, Is greater than, Is above, Is higher <, Is less than, Is below, Is lower than,
than, Is longer than, Is bigger than, Is Is shorter than, Is smaller than, Is
increased decreased or reduced from
=, Is equal to, Is the same as, Has not ≠, Is not equal to, Is not the same as,
changed from, Is no different from, Is Has changed from, Is different from, Is
not

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The null and the alternative hypotheses are stated together as shown (where k
represents a specified number).
Two-tailed test Right-tailed test Left-tailed test
H o : μ=k H o : μ=k H o : μ=k
H a : μ≠ k H a : μ>k H a : μ<k

The null hypothesis contains the equals sign. This is done because in most
professional journals, and when we test the null hypothesis, the assumption is that the
mean, proportion, or standard deviation is equal to a given specific value. Also when
a researcher conducts a study, he or she is generally looking for evidence to support
the claim. Therefore, the claim should be stated as the alternative hypothesis, i.e.,
using > or < or ≠. Because of this, the alternative hypothesis is sometimes called the
research hypothesis.

A claim, though, can be stated as either the null hypothesis or the alternative
hypothesis; however, the statistical evidence can only support the claim if it is the
alternative hypothesis. Statistical hypothesis can be used to reject the claim if the
claim is the null hypothesis. These facts are important when you are stating the
conclusion of a statistical study.

Before we go any further, consider the following examples.

1. A medical researcher is interested in finding out whether a new medication will
have any undesirable side effects. The researcher is particularly concerned with
the pulse rate of the patients who take the medication. Will the pulse rate increase,
decrease, or remain unchanged after a patient takes a medication?
2. A chemist invents an additive to increase the life of an automobile battery. The
mean lifetime of the automobile battery without the additive is 36 months.
3. A contractor wishes to lower heating bills by using a special type of insulation in
houses. The average of the monthly heating bills is $78.
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Solution

1. Since the researcher knows that the mean pulse rate for the population under
study is 82 beats per minute, the hypotheses for this situation are
H o : μ=82 H a : μ≠ 82

The null hypothesis states that the mean will remain unchanged, and the
alternative hypothesis states that it will be different. This test is called a two-
tailed test, since the possible side effects of the medicine could be to raise or
lower the pulse rate.

Definition 22: In a two-tailed test, the null hypothesis should be rejected when
the test value is in either of the two critical regions.

2. The corresponding hypotheses are

H o : μ=36 H a : μ>36

In this situation, the chemist is interested only in increasing the lifetime of the
batteries, so her alternative hypothesis is that the mean is greater than 36 months.
The null hypothesis is that the mean is equal to 36 months. This test is called a
right-tailed test, since the interest is in the increase only.

3. The corresponding hypotheses are

H o : μ=$ 78 H a : μ< $ 78

This is a left-tailed test, since the contractor is interested only in lowering heating
costs.

Definition 23: A one-tailed test indicates that the null hypothesis should be
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rejected when the test value is in the critical region on one side of the mean. A
one-tailed test is either a right-tailed or a left-tailed, depending on the
direction of the inequality of the alternative hypothesis.

More illustrations:
1. Research Topic: “ A Study on the Performance of Men and Women in the
Licensure Examination for Engineers”
 A two-tailed (non-directional) way of stating the statistical hypotheses
Ho: There is no difference in the performance of men and women in the
licensure examination for engineers. ( μm =μ w )
Ha: There is difference in the performance of men and women in the
licensure examination for engineers. ( μm ≠ μ w)
 A one-tailed (directional) way of stating the statistical hypotheses
Ho: Women perform equally well as men in the licensure examination for
engineers. ( μw =μ m )
Ha: Women perform better in the licensure examination for engineers than
men. ( μw > μm )

2. A consumer protection watchdog wants to test a company’s claim that their

carton of UHT milk contains on the average a net weight of 1.2 liters.
 Non-directional
Ho: The mean content of a carton milk is 1.2. ( μ=1.2)
Ha: The mean content of a carton milk is not 1.2. ( μ ≠1.2)
 Directional
Ho: The mean content of a carton milk is 1.2. ( μ=1.2)
Ha: The mean content of a carton milk is less than 1.2. ( μ<1.2)

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After stating the hypothesis, the researcher designs the study. The
researcher selects the correct statistical test, chooses an appropriate level of
significance, and formulates a plan for conducting the study.

Definition 24: A statistical test uses the data obtained from a sample to make a
decision about whether the null hypothesis should be rejected.

The numerical value obtained from a statistical test is called the test value.

In a statistical test, the mean is computed for the data obtained from the
sample and is compared with the population mean. Then a decision is made to
reject or not reject the null hypothesis on the basis of the value obtained from the
statistical test. If the difference is significant, the null hypothesis is rejected. If the
difference is not significant, then the null hypothesis is not rejected.
In a hypothesis-testing situation, there are four possible outcomes. In
reality, the null hypothesis may or may not be true, and a decision is made to
reject or not reject it on the basis of the data obtained from a sample, as shown in
the following illustration. Notice that there are two possibilities for a correct
decision and two possibilities for an incorrect decision.

H o true H o false

Reject H o Error Correct decision

Type I

Do not reject H o Correct decision Error

Type II

Remarks: It is often argued that our only two choices are to (1) reject or (2) not
reject the null hypothesis. It is never possible to prove beyond any doubt that the
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null hypothesis is correct, so it is never possible to “accept” it as such. If we were

to hypothesize that μ is 115, and then actually found an X exactly equal to 115, it
would still not prove that μ is 115. It could well be that μ is actually 113, and our
sample just happened to contain a few of the larger observations. Thus, even a
sample mean exactly equal to our hypothesized value does not conclusively prove
that the null hypothesis is correct.

Definition 25: A type I error occurs if you reject the null hypothesis when it
is true. A type II error occurs if you do not reject the null hypothesis when it
is false.

Definition 26: The level of significance is the maximum probability of

committing a type I error. This probability is symbolized by α (Greek letter
alpha). That is, P ( type I error )=α .

The probability of a type II error is symbolized by β , the Greek letter beta.

That is, P ( type II error )= β. In most hypothesis-testing situations, β cannot be
easily computed; however, α and β are related in that decreasing one increases
the other.

Statisticians generally agree on using three arbitrary significance levels:

the 0.10, 0.05, and 0.01 levels. That is, if the null hypothesis is rejected, the
probability of a type I error will be 10%, 5%, or 1%, depending on which level of
significance is used. Here is another way of putting it: When α =0.10 , there is a
10% chance of rejecting a true null hypothesis; when α =0.05 , there is a 5%
chance of rejecting a true null hypothesis; when α =0.01 , there is a 1% chance of
rejecting a true null hypothesis.

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In a hypothesis-testing situation, the researcher decides what level of

significance to use. It does not have to be the 0.10, 0.05, or 0.01 level. It can be
any level, depending on the seriousness of the type I error. Once the α value is
specified, the β value is fixed for any given sample size. The β value, or the
probability of a type II error, depends upon the true value of μ. If μ is, say 100, a
particular value for β will exist. However, if μ is really 110, a different value for
β exists.
After a significance level is chosen, a critical value is selected for the
appropriate test.

Definition 27: The critical value separates the critical region from the
noncritical region. The symbol for critical value is c.v.
The critical or rejection region is the range of values of the test value that
indicates that there is a significant difference and that the null hypothesis
should be rejected.
The noncritical or nonrejection region is the range of values of the test value
that indicates that there is no significant difference (the difference was probably
due to chance) and that the null hypothesis should not be rejected.

The procedure in performing hypothesis testing follows ( The five-step solution):

1. State the null and alternative hypothesis.
2. State the decision rule.
3. Indicate the appropriate test and level of significance to be used.
4. Show computation for tabulated and critical/p-value.
5. Make a decision and conclusion.
(Quinto 2017, Gonzales and Nocon, 2013)

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III. SUGGESTED REFERENCES

Bluman, A.G. Elementary Statistics: A Step-by-step Approach, 2017.

Bohnenblust, S., and Kuzma, J., Statistics for the Health Sciences , 5th edition,
McGraw Hill, 2005.

Gonzales, J., and Nocon, R., Essential Statistics , Mutya Publishing, 2013.

Quinto, Glenda.Coursepack in Statistics. College of Math and Engineering,Ateneo de

Naga University.

Pagoso, et al., Fundamental Statistics for College Students

Parel, E., and Alonzo, A., Introduction to Statistical Methods, 5th edition.

Reyes, F., Applied Basic Statistics , 3rd edition.

Walpole, R., Introduction to Statistics, 3rd edition .,Prentice Hall, 1998.

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