INEN 201: STATISTICAL ANALYSIS FOR INDUSTRIAL ENGINEERING 2
INFERENTIAL STATISTICS – HYPOTHESIS TESTING
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OUTLINE
Inferential Statistics
Hypothesis Testing
What is Hypothesis?
The Null and Alternative Hypothesis
One-tailed and Two-tailed Tests
Critical Value
Statistical Significance
Type I Error and Type II Error
Central Limit Theorem
Steps in Hypothesis Testing
Critical Value Using Z-Test
Selecting the Appropriate Statistical Test
One Sample Mean Test (Z-Test)
One Sample Mean Test (T-Test)
Simple Tests of Hypothesis
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A Hypothesis is a tentative explanation for
certain events, phenomena, and behavior.
●
A Hypothesis is a statement of prediction of the
relationship between or among variables.
●
A Hypothesis is a statement about the concept
that refers to observable phenomena which
may be judge as true or false and is subject to
empirical testing.
●
A Hypothesis is testable which means that the
relationship between these variables can be
put into test by means of the application of
appropriate statistical test on the data
gathered about variables.
●
A Hypothesis a statement that require us
(Industrial Engineers) to decide whether to
accept or reject about some statistical
parameter.
●
INFERENTIAL STATISTICS
●
The theory of statistical inference consists of
methods by which one makes inference or
generalizations about a population.
TWO MAJOR AREAS OF STATISTICAL INFERENCE
1.
2.
●
Point Estimate: A single value used to
estimate a population parameter.
A Hypothesis is a statement or tentative theory
or assertion or conjecture which aims to explain
facts about the real world.
●
Interval Estimate: A range of values used
to estimate a population parameter.
TWO TYPES OF STATISTICAL HYPOTHESES
ESTIMATION OF PARAMETERS
TESTS OF HYPOTHESIS
●
1.
Testing assumptions or claims about a
population parameter based on sample
data.
NULL HYPOTHESIS (𝐇𝟎 )
●
A null hypothesis must always express
the idea of non-significance of
difference or the idea of no relationship.
●
It is the hypothesis to be tested and it
represents what the investigation
doubts to be true.
●
It is the hypothesis that we hope to
reject.
●
It is the starting point of the testing
process.
HYPOTHESIS TESTING
●
●
Engr. Romeo Ilagan|1|PRELIM
Hypothesis testing is the process of
determining the probability or acceptability
of a hypothesis as derived from a theory
through scientific data collection and through
application of appropriate statistical test.
This is the decision-making procedure about
the hypothesis.
2.
ALTERNATIVE HYPOTHESIS (𝐇𝟏 )
●
An alternative hypothesis is the opposite
of the null hypothesis.
●
It generally represents the hypothetical
statement that the researcher wants to
prove or claim the manufacturer eager to
substantiate.
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It is the operational statement of the
theory that the experimenter or
researcher believe to be true and wishes
to prove.
●
Alternative hypotheses are classified as
either non-directional (two-tailed) or
directional hypotheses (one-tailed).
PURPOSE OF STATISTICAL HYPOTHESIS TEST
●
To permit generalizations (about population
parameters) from a sample to the population
which it came.
●
This process involves:
○
assumptions about the population
○
the use of probabilities to estimate the
likelihood of the results obtained in the
sample
○
a random sample has been properly selected
WHAT IS HYPOTHESIS?
●
A Hypothesis is basically a statement about the
target population. This is formulated as a result
of years of observation and researches.
INFERENTIAL STATISTICS – HYPOTHESIS TESTING
Non-directional (two-tailed): Suggests that there is
a difference but does not specify the direction of the
effect.
Directional (one-tailed): Indicates the expected
direction of the effect (e.g., an increase or decrease).
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ALTERNATIVE HYPOTHESES
A.
B.
ONE TAILED AND TWO TAILED TESTS
NON-DIRECTIONAL HYPOTHESIS (𝐇𝒂 )
ONE-TAILED TEST (DIRECTIONAL)
●
One which asserts that one value is
different from another (or others).
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The rejection region which is equal to alpha (α
) is a place on side of the distribution.
●
It is an assertion that there is a
significant difference or significant
relationship between two statistical
measures (or among three or more
summary measures).
●
A one tailed test has one critical value, either
positive or negative.
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It implies non-equality (≠).
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Non-directional hypotheses are
called two-sided hypotheses.
TWO-TAILED TEST (NON-DIRECTIONAL)
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The two tails of the distribution are considered
the rejection region and a value of alpha (α) is
equally divided into these two tailed.
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A two tailed test has two critical values, one
positive and the other negative.
●
The critical value of the test statistics is the
value that divides the distribution of sample
means into rejection and acceptance regions.
●
It corresponds to the vertical line in the above
figure.
●
A
test
which
involves
non-directional
alternative hypothesis has two shaded regions.
also
DIRECTIONAL HYPOTHESIS (𝐇𝟏 )
●
It is sometimes
hypothesis.
called
predictive
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It is an assertion that one measure is less
than (or greater than) another measure
of similar nature.
●
A directional hypothesis involves one of
the order relations, “less than” (<) or
“greater than” (>).
●
Directional hypotheses are also known as
one-sided hypotheses.
THE NULL AND ALTERNATIVE HYPOTHESIS
CRITICAL VALUE
DECISION TO REJECT OR TO ACCEPT
THE NULL HYPOTHESIS
THE NULL AND ALTERNATIVE HYPOTHESIS
●
If the null hypothesis is rejected as a result of
the sample evidence, then the alternative
hypothesis is the conclusion.
●
If there is no sufficient evidence to reject the
null hypothesis, it is retained, but not
accepted.
○
In this case, the null hypothesis is not necessarily
true, but it just cannot be rejected from the
current evidence from the sample data.
INFERENTIAL STATISTICS – HYPOTHESIS TESTING
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When the computed value of the test statistic
falls within the acceptance region, the null
hypothesis is accepted while if it falls within the
rejection region, the null hypothesis is
rejected.
●
When the absolute value of its critical value test
statistic is less than or equal the absolute
value of its critical value, accept the null
hypothesis; otherwise, reject the null
hypothesis
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STATISTICAL SIGNIFICANCE
●
The difference in the sample means may exist
for one of two reasons:
1.
NO ACTUAL DIFFERENCE (CHANCE)
●
2.
There is no actual difference between the
groups because both samples were
taken from the same population – the
observed difference is just a chance
occurrence due to the error involved in
the sampling.
CENTRAL LIMIT THEOREM
●
It states that as the sample size (n) increases,
the distribution of the sample means
approaches a normal distribution, regardless
of the original shape of the population
distribution (skewed, flat, normal, or any other
shape).
●
The distribution of the sample means will
become more normally distributed as n
becomes large.
IMPORTANCE
A REAL DIFFERENCE EXISTS
●
Difference actually exists because each
sample
came
from
a
different
population and the difference is
therefore real.
●
When the sample is sufficiently large, you can
assume that the distribution of sample means is
normal.
●
It is therefore easy to compare the samples you
have collected to the theoretical distribution of
sample means
Null Hypothesis: Claims that no real difference exists
between the sample means (any observed difference is
due to chance).
Alternative Hypothesis: Asserts that a real difference
exists between the sample means.
FACTORS INFLUENCING
STATISTICAL SIGNIFICANCE
●
the number of samples
●
the variances of the population and the sample
●
the size of the risk (confidence) taken by the
researches that their conclusion will be wrong
STEPS IN HYPOTHESIS TESTING
1.
2.
3.
4.
5.
6.
TYPE I ERROR AND TYPE II ERROR
●
Data contained in published and unpublished
documents, reports, statistics, manuscripts,
letters, diaries.
State the null hypothesis.
Select an appropriate alternative hypothesis
(H1 ).
Choose the appropriate statistical test.
Select the desired level of significance to be
used. The most common level is 0.05, but 0.01
is also widely used. Sometimes, other alpha(α)
levels such as 0.001, 0.025, or 0.10 are also used.
Compute the calculated value and determine
the critical test value.
Make the decision. REJECT the null hypothesis
if the calculated value is greater than the
critical value, otherwise “do not reject the null
hypothesis”.
CRITICAL VALUE USING Z-TEST
TYPE I ERROR
●
Sometimes called alpha error (α).
●
If the researcher decides to reject the null
hypothesis when it is actually true.
TYPE II ERROR
●
sometimes called beta error (𝜷).
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If the researcher fails to reject the null
hypothesis when it is actually false.
SELECTING THE APPROPRIATE STATISTICAL TEST
Z-TEST
LEVEL OF SIGNIFICANCE (𝜶)
●
It is the probability of committing a type 1 error.
●
This is the value of 1-α.
LEVEL OF CONFIDENCE (𝜷)
INFERENTIAL STATISTICS – HYPOTHESIS TESTING
●
If the sample size is large (n ≥ 30).
●
When the population standard deviation is
known.
●
If the population standard deviation is
unknown, a z-test may still be used for large
samples (n≥30) due to the Central Limit
Theorem.
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INEN 201: STATISTICAL ANALYSIS FOR INDUSTRIAL ENGINEERING 2
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T-TEST
●
If the sample size is small (n < 30).
●
When the population standard deviation is
unknown.
ONE TAILED OR TWO-TAILED TEST
●
One-tailed test: Used when the alternative
hypothesis is directional, meaning it predicts a
specific direction of effect.
●
Two-tailed test: Used when the alternative
hypothesis is non-directional, meaning it does
not predict a specific direction of effect.
●
The one-sample mean z-test is used to
compare a sample mean (X) to a known
population mean (µ) to determine if there is a
significant difference between them.
ONE SAMPLE MEAN TEST (Z-TEST)
𝒁=
(𝑿 − 𝝁) • (√𝒏)
𝝈
INFERENTIAL STATISTICS – HYPOTHESIS TESTING
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INEN 201: STATISTICAL ANALYSIS FOR INDUSTRIAL ENGINEERING 2
ONE SAMPLE MEAN TEST (T-TEST)
●
Engr. Romeo Ilagan|1|PRELIM
SIMPLE TESTS OF HYPOTHESIS
The one-sample t-test is used for testing a
hypothesis about a single mean when the
sample size is small (n < 30) and the
population
standard
deviation
(σ)
is
unknown.
𝒕=
(𝑿 − 𝝁) • (√𝒏)
𝒔
Exercises:
INFERENTIAL STATISTICS – HYPOTHESIS TESTING
BSIE 2-1 | ALJANE
INEN 201: STATISTICAL ANALYSIS FOR INDUSTRIAL ENGINEERING 2
INFERENTIAL STATISTICS – HYPOTHESIS TESTING
Engr. Romeo Ilagan|1|PRELIM
BSIE 2-1 | ALJANE