Transcript two-tailed test - Bakersfield College

Chapter 8
Introduction to Hypothesis Testing
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Name of the game…

Hypothesis testing

Statistical method that uses sample data to
evaluate a hypothesis about a population
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Experimental Hypotheses
Experimental hypotheses describe the
predicted outcome we may or may not find in
an experiment.
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Order of Procedure




State hyp about pop
Before selecting a sample..use hyp to
predict characteristics that sample should
have
Obtain random sample
Compare sample data w/ prediction made
in hyp
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Of interest to the researcher…


Did the treatment have any effect on the
individuals
Must be large (significant) differences in
means
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New Statistical Notation

The symbol for greater than is >.

The symbol for less than is <.

The symbol for greater than or equal to
is ≥.

The symbol for less than or equal to
is ≤.

The symbol for not equal to is ≠.
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The Role of Inferential
Statistics in Research
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Sampling Error
Remember:
Sampling error results when random chance
produces a sample statistic that does not
equal the population parameter it represents.
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Setting up Inferential Procedures
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Null Hypothesis H0
The null hypothesis describes the
population parameters that the sample data
represent if the predicted relationship does
not exist.
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Alternative Hypothesis H1
The alternative hypothesis describes the
population parameters that the sample data
represent if the predicted relationship exists.
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A Graph Showing the Existence of
a Relationship
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A Graph Showing That a Relationship
Does Not Exist
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Interpreting Significant Results

When we reject H0 and accept H1, we do
not prove that H0 is false

While it is unlikely for a mean that lies
within the rejection region to occur, the
sampling distribution shows that such
means do occur once in a while
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Failing to Reject H0

When the statistic does not fall beyond the critical
value, the statistic does not lie within the region of
rejection, so we do not reject H0

When we fail to reject H0 we say the results are
nonsignificant. Nonsignificant indicates that the
results are likely to occur if the predicted
relationship does not exist in the population.
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Interpreting Nonsignificant Results

When we fail to reject H0, we do not prove
that H0 is true

Nonsignificant results provide no
convincing evidence—one way or the
other—as to whether a relationship exists
in nature
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Errors in Statistical
Decision Making
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Type I Errors

A Type I error is defined as rejecting H0 when H0
is true

In a Type I error, there is so much sampling error
that we conclude that the predicted relationship
exists when it really does not

The theoretical probability of a Type I error equals
a
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Alpha a


Probability that the test will lead to a Type I
error
Alpha level determines the probability of
obtaining sample data in the critical region
even though the null hypo is true
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Type II Errors

A Type II error is defined as retaining H0 when H0
is false (and H1 is true)

In a Type II error, the sample mean is so close to
the m described by H0 that we conclude that the
predicted relationship does not exist when it really
does

The probability of a Type II error is b
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Power

The goal of research is to reject H0 when
H0 is false

The probability of rejecting H0 when it is
false is called power
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Possible Results of Rejecting or
Retaining H0
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Parametric Statistics

Parametric statistics are procedures that require
certain assumptions about the characteristics of
the populations being represented. Two
assumptions are common to all parametric
procedures:
The population of dependent scores forms a normal
distribution
and
 The scores are interval or ratio.

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Nonparametric Procedures

Nonparametric statistics are inferential
procedures that do not require stringent
assumptions about the populations being
represented.
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Robust Procedures
Parametric procedures are robust. If the
data don’t meet the assumptions of the
procedure perfectly, we will have only a
negligible amount of error in the inferences
we draw.
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Predicting a Relationship

A two-tailed test is used when we predict
that there is a relationship, but do not
predict the direction in which scores will
change.

A one-tailed test is used when we predict
the direction in which scores will change.
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The One-Tailed Test
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One-Tailed Hypotheses

In a one-tailed test, if it is hypothesized that the independent
variable causes an increase in scores, then the null
hypothesis is that the population mean is less than or equal
to a given value and the alternative hypothesis is that the
population mean is greater than the same value. For
example:


H0: m ≤ 50
Ha: m > 50
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A Sampling Distribution Showing the
Region of Rejection for a One-tailed Test of
Whether Scores Increase
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One-Tailed Hypotheses

In a one-tailed test, if it is hypothesized that the independent
variable causes a decrease in scores, then the null
hypothesis is that the population mean is greater than or
equal to a given value and the alternative hypothesis is that
the population mean is less than the same value. For
example:


H0: m ≥ 50
Ha: m < 50
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A Sampling Distribution Showing the Region of
Rejection for a One-tailed Test of Whether Scores
Decrease
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Choosing One-Tailed Versus
Two-Tailed Tests
Use a one-tailed test only when confident of
the direction in which the dependent variable
scores will change. When in doubt, use a
two-tailed test.
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Performing the z-Test
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The z-Test
The z-test is the procedure for computing a
z-score for a sample mean on the sampling
distribution of means.
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Assumptions of the z-Test
1.
We have randomly selected one sample
2.
The dependent variable is at least approximately normally
distributed in the population and involves an interval or ratio
scale
3.
We know the mean of the population of raw scores under
some other condition of the independent variable
4. We know the true standard deviation of the population
( X ) described by the null hypothesis
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Setting up for a Two-Tailed Test
1.
Choose alpha. Common values are 0.05
and 0.01.
2.
Locate the region of rejection. For a twotailed test, this will involve defining an area
in both tails of the sampling distribution.
3.
Determine the critical value. Using the
chosen alpha, find the zcrit value that gives
the appropriate region of rejection.
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Showing
the Region of Rejection for a = 0.05 in
a Two-tailed Test
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Two-Tailed Hypotheses

In a two-tailed test, the null hypothesis
states that the population mean equals a
given value. For example, H0: m = 100.

In a two-tailed test, the alternative
hypothesis states that the population mean
does not equal the same given value as in
the null hypothesis. For example, Ha: m 
100.
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Computing z
• The z-score is computed using the same
formula as before
zobt 
X m
where
X 
X
X
N
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Rejecting H0

When the zobt falls beyond the critical value,
the statistic lies in the region of rejection, so
we reject H0 and accept Ha

When we reject H0 and accept Ha we say
the results are significant. Significant
indicates that the results are too unlikely to
occur if the predicted relationship does not
exist in the population.
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Interpreting Significant Results

When we reject H0 and accept Ha, we do
not prove that H0 is false

While it is unlikely for a mean that lies
within the rejection region to occur, the
sampling distribution shows that such
means do occur once in a while
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Failing to Reject H0

When the zobt does not fall beyond the
critical value, the statistic does not lie within
the region of rejection, so we do not reject
H0

When we fail to reject H0 we say the results
are nonsignificant. Nonsignificant
indicates that the results are likely to occur
if the predicted relationship does not exist
in the population.
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Interpreting Nonsignificant Results

When we fail to reject H0, we do not prove
that H0 is true

Nonsignificant results provide no
convincing evidence—one way or the
other—as to whether a relationship exists
in nature
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Summary of the z-Test
Determine the experimental hypotheses
and create the statistical hypothesis
2.Compute X and compute zobt
1.
3.Set up the sampling distribution
4.Compare zobt to zcrit
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