Hypothesis Testing
Researchers are interested in in answering many types of questions. For example, a scientist might want
to know whether the earth is warming up. A physician might want to find out whether a new medicine will
lower a person’s blood pressure. An educator might wish to see whether a new teaching technique is better
than a traditional one. These questions can be addressed by through statistical hypothesis testing.
Hypothesis testing is a decision-making for evaluating claims about a population. Every hypothesis-testing
situation begins with a hypothesis.
A hypothesis is an assumption about something.
Statistical hypothesis is a conjecture about a population parameter. This conjecture may or may not be true.
Types of statistical hypothesis:
1. Null hypothesis (symbolized by Ho) is a statistical hypothesis that states that there is no difference
between a parameter and a specific value or that there is no difference between parameters. Ho is
always stated as an equality claim involving parameters. It is subjected to statistical test.
2. Alternative hypothesis (symbolized by H1or Ha) is a statistical hypothesis that states the existence of
a difference between a parameter and a specific value, or states that there is a difference between
two parameters. It is an inequality claim that contradicts Ho.
The alternative hypothesis states that an observed difference is likely to be genuine and not likely to
have occurred by chance alone. It may be one-side (using either > or <) or two-sided (using ≠ ).
Example 1. A medical researcher is interested in finding out whether a new medication will have any
undesirable 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
the medication?
Since the researcher knows that the mean pulse rate for the population under study is 82 beats
per minute, the hypothesis are:
Ho: 𝜇 = 82
(The mean pulse rate is 82 beats per minute.)
Ha: 𝜇 ≠ 82
(The mean pulse rate is not 82 beats per minute.)
The null hypothesis specifies 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 medication could be to lower or raise the pulse rate.
Example 2. A chemist invents an additive to increase the life of an automobile battery. If the mean life of the
automobile battery is 36 months, then the hypotheses are:
Ho: 𝜇= 36
Ha: 𝜇 > 36
The chemist is interested only in increasing the lifetime of the batteries so the alternative
hypothesis is that the mean is greater than 36. The null hypothesis is that the mean is equal to 36
months. This test is one-tailed specifically right-tailed, since the interest is in an increase only.
Example 3. A contractor wishes to lower heating bills by using a special type of insulation in houses. If the
average of the monthly heating bills is Php 3900.00, the hypotheses about heating costs with the use of
insulation are
Ho: 𝜇 = 3900
Ha: 𝜇 < 3900
This test is left-tailed test, since the contractor is interested only in lowering heating costs.
Example 4.
Objective: To determine if there is a significant difference in the mean score of students when they are
classified as to sex.
Ho: There is no significant difference in the mean score of students when they are classified as to sex.
(mean of males = mean of females)
Ha: There is a significant difference in the mean score of students when they are classified as to sex.
(mean of males ≠ mean of females)
Example 5.
Objective: To determine if smoking practices of students are significantly associated to the smoking practices of
their parents.
Ho: The smoking practices of students are not significantly associated to the smoking practices of their parents
(Ho: r = 0).
Ha: The smoking practices of students are significantly associated to the smoking practices of their parents
(Ho: r ≠ 0).
Example 6.
Objective: To determine if there is a significant relationship between the age of students and the number of
sticks of cigarettes they smoked.
Ho: There is no significant relationship between the age of students and the number of sticks of cigarettes they
smoked(Ho: r = 0).
Ha: There is a significant relationship between the age of students and the number of sticks of cigarettes they
smoked (Ho: r ≠ 0).
Directional vs. Nondirectional Hypothesis
Sometimes called a two-tailed test, a test of a nondirectional alternative hypothesis does not state the
direction of the difference; it indicates only that a difference exists. In contrast, a directional alternative
hypothesis (one-tailed test) specifies the direction of the tested relationship or difference.
Examples:
Two-tailed
Ho: There is no significant difference between the mean score of males and the mean score of
females.
Ha: There is a significant difference between the mean score of males and the mean score of
females.
One-tailed
Ho: There is no significant difference between the mean score of males and the mean score of
females.
Ha: The mean score of males is significantly higher than the mean score of females.
Activity. State the null and alternative hypothesis for each conjecture.
1. A researcher thinks that if expectant mothers take vitamin pills, the birth weight of the babies will
increase. The average birth weight of the population is 8.6 pounds.
2. An engineer hypothesizes that the mean number of defects can be decreased in a manufacturing
process of compact disks by using robots instead of human for certain tasks. The mean number of
defective disks per 1000 is 18.
3. A psychologist feels that playing soft music during a test will change the results of the test. The
psychologist is not sure whether the grades will be higher or lower. In the past, the mean of the scores
was 73.
Type I and Type II Errors
Type I error, also known as a “false positive” is the error of rejecting a null hypothesis when it is
actually true.
Type II error, also known as a "false negative" is the error of not rejecting a null hypothesis when the
alternative hypothesis is the true state of nature.
Ho true
Ho false
Reject
Ho
Error
Type I
Correct
Decision
Do not
reject
Ho
Correct
decision
Error
Type II
Level of Significance of a test
In hypothesis testing, the significance level is the criterion used for rejecting the null hypothesis.It is
often expressed as the so-called p-value.
The level of significance of a test, denoted by𝛼, is the probability of committing a Type I error or the
probability of rejecting a true Ho.
Test of Hypothesis
A test of hypotheses is a method for using sample data to decide whether the null hypothesis should
be rejected. It is a procedure for deciding if a null hypothesis should be accepted or rejected in favor of
an alternative hypothesis.
Hypothesis testing is about testing to see whether the stated hypothesis is acceptable or not.
Steps in Hypothesis Testing
1. State the null and alternative hypothesis
2.
3.
4.
5.
6.
Set the level of significance
Determine the test to be used
Determine the tabular value for the test
Compute for the value of the test
State the statistical and research conclusion
GENERAL CRITERIA FOR TEST SELECTION
1. Level of measurementachieved by the variable
The choice of statistical test depends upon the type of data the researcher has at hand, i.e.,
whether the observations are in the nominal, ordinal, interval or ratio scale of measurement.
2. Objectives/purposes of the study
The research question or hypothesis is typically phrased in terms of finding differences,
relationships, or predicting.
3. Design of the study
The type of statistical test to employ is dependent on whether the samples are independent or
paired.
4. Assumptions of the test
In general, all statistical tests of hypothesis can be classified as parametric or non-parametric.
These two main categories are differentiated by the conditions or assumptions each specifies and
requires for its validity.
Tests of significance: Two independent samples
Level of measurement
Test of significance
Nominal
•
chi-square test of independence for a
cross tabulation
Ordinal
•
Mann-Whitney U test or Wilcoxon W test
Interval/Ratio
•
t-test for the equality of two means
Tests of significance: Two dependent samples
Level of measurement
Test of significance
Nominal
•
McNemar test for change
Ordinal
•
Wilcoxon matched-pairs signed-ranks test
Interval/Ratio
•
t-test for the mean difference
Tests of significance: More than two independent samples
Level of measurement
Test of significance
Nominal
•
Chi-square test for independence
Ordinal
•
Kruskal Wallis H test
Interval/Ratio
•
ANOVA F-test for the equality of means
Tests of Relationship
Level of measurement
Test of significance
Nominal
•
Chi-square test for independence
Ordinal
•
Spearman Rho
Interval/Ratio
•
Pearson r