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Hypothesis Testing
©
Hypothesis
An idea or explanation for something that
is based on known facts but not yet has
been proved
Formulating the Hypothesis
The null hypothesis is a statement about
the population value that will be tested.
The null hypothesis will be rejected only if
the sample data provide substantial
contradictory evidence.
Formulating the Hypothesis
The alternative hypothesis is the
hypothesis that includes all population
values not covered by the null hypothesis.
The alternative hypothesis is deemed to be
true if the null hypothesis is rejected.
Formulating the Hypothesis
The research hypothesis is the hypothesis
the decision maker attempts to
demonstrate to be true. Since this is the
hypothesis deemed to be the most
important to the decision maker, it will not
be declared true unless the sample data
strongly indicates that it is true.
Types of Statistical Errors
 Type I Error - This type of statistical
error occurs when the null hypothesis
is true and is rejected.
 Type II Error - This type of statistical
error occurs when the null hypothesis
is false and is not rejected.
 The power of a test is the probability (1 - ) of
rejecting the null hypothesis when it is false and should
be rejected.
 Although is unknown, it is related to . An extremely
low value of (e.g., = 0.001) will result in intolerably
high errors.
 So it is necessary to balance the two types of errors.
Power of a Test










Establishing the Decision Rule
The critical value is the value of a statistic
corresponding to a given significance level.
This cutoff value determines the boundary
between the samples resulting in a test
statistic that leads to rejecting the null
hypothesis and those that lead to a decision
not to reject the null hypothesis.
Establishing the Decision Rule
The significance level is the maximum
probability of committing a Type I statistical
error. The probability is denoted by the
symbol .
Reject H0

x
x
25


Do not reject H0
Sampling Distribution
Maximum probability
of committing a Type I
error = 
Establishing the Decision Rule

x
z
25


Rejection region
 = 0.10
28
.
1


z
0
From the standard normal table
28
.
1
10
.
0 


z
Then
28
.
1


z
0.5 0.4
Establishing the Critical Value
as a z -Value
?


x
z
25


Rejection region
 = 0.10
28
.
1


z
0
0.5 0.4
Example of Determining the
Critical Value
64
3


n
x



x
for
Solving
64
3
28
.
1
25



n
z
x



48
.
25


x
Establishing the Decision Rule
The test statistic is a function of the
sampled observations that provides a
basis for testing a statistical
hypothesis.
Summary of Hypothesis
Testing Process
The hypothesis testing process can be summarized in 6
steps:
 Determine the null hypothesis and the alternative
hypothesis.
 Determine the desired significance level ().
 Define the test method and sample size and
determine a critical value.
 Select the sample, calculate sample mean, and
calculate the z-value or p-value.
 Establish a decision rule comparing the sample
statistic with the critical value.
 Reach a conclusion regarding the null hypothesis.
One-Tailed Hypothesis Tests
A one-tailed hypothesis test is a
test in which the entire rejection
region is located in one tail of the
test statistic’s distribution.
Two-Tailed Hypothesis Tests
A two-tailed hypothesis test is a
test in which the rejection region
is split between the two tails of
the test statistic’s distribution.
Hypothesis Tests for Two
Population Variances
HYPOTHESIS TESTING STEPS
 Formulate the null and alternative hypotheses in
terms of the population parameter of interest.
 Determine the level of significance.
 Determine the critical value of the test statistic.
 Select the sample and compute the test
statistic.
 Compare the calculated test statistic to the
critical value and reach a conclusion.
Independent Samples
Independent samples are those samples
selected from two or more populations in
such a way that the occurrence of values in
one sample have no influence on the
probability of the occurrence of values in
the other sample(s).
 Thank you

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312320.pptx

  • 2. Hypothesis An idea or explanation for something that is based on known facts but not yet has been proved
  • 3. Formulating the Hypothesis The null hypothesis is a statement about the population value that will be tested. The null hypothesis will be rejected only if the sample data provide substantial contradictory evidence.
  • 4. Formulating the Hypothesis The alternative hypothesis is the hypothesis that includes all population values not covered by the null hypothesis. The alternative hypothesis is deemed to be true if the null hypothesis is rejected.
  • 5. Formulating the Hypothesis The research hypothesis is the hypothesis the decision maker attempts to demonstrate to be true. Since this is the hypothesis deemed to be the most important to the decision maker, it will not be declared true unless the sample data strongly indicates that it is true.
  • 6. Types of Statistical Errors  Type I Error - This type of statistical error occurs when the null hypothesis is true and is rejected.  Type II Error - This type of statistical error occurs when the null hypothesis is false and is not rejected.
  • 7.  The power of a test is the probability (1 - ) of rejecting the null hypothesis when it is false and should be rejected.  Although is unknown, it is related to . An extremely low value of (e.g., = 0.001) will result in intolerably high errors.  So it is necessary to balance the two types of errors. Power of a Test          
  • 8. Establishing the Decision Rule The critical value is the value of a statistic corresponding to a given significance level. This cutoff value determines the boundary between the samples resulting in a test statistic that leads to rejecting the null hypothesis and those that lead to a decision not to reject the null hypothesis.
  • 9. Establishing the Decision Rule The significance level is the maximum probability of committing a Type I statistical error. The probability is denoted by the symbol .
  • 10. Reject H0  x x 25   Do not reject H0 Sampling Distribution Maximum probability of committing a Type I error =  Establishing the Decision Rule
  • 11.  x z 25   Rejection region  = 0.10 28 . 1   z 0 From the standard normal table 28 . 1 10 . 0    z Then 28 . 1   z 0.5 0.4 Establishing the Critical Value as a z -Value
  • 12. ?   x z 25   Rejection region  = 0.10 28 . 1   z 0 0.5 0.4 Example of Determining the Critical Value 64 3   n x    x for Solving 64 3 28 . 1 25    n z x    48 . 25   x
  • 13. Establishing the Decision Rule The test statistic is a function of the sampled observations that provides a basis for testing a statistical hypothesis.
  • 14. Summary of Hypothesis Testing Process The hypothesis testing process can be summarized in 6 steps:  Determine the null hypothesis and the alternative hypothesis.  Determine the desired significance level ().  Define the test method and sample size and determine a critical value.  Select the sample, calculate sample mean, and calculate the z-value or p-value.  Establish a decision rule comparing the sample statistic with the critical value.  Reach a conclusion regarding the null hypothesis.
  • 15. One-Tailed Hypothesis Tests A one-tailed hypothesis test is a test in which the entire rejection region is located in one tail of the test statistic’s distribution.
  • 16. Two-Tailed Hypothesis Tests A two-tailed hypothesis test is a test in which the rejection region is split between the two tails of the test statistic’s distribution.
  • 17. Hypothesis Tests for Two Population Variances HYPOTHESIS TESTING STEPS  Formulate the null and alternative hypotheses in terms of the population parameter of interest.  Determine the level of significance.  Determine the critical value of the test statistic.  Select the sample and compute the test statistic.  Compare the calculated test statistic to the critical value and reach a conclusion.
  • 18. Independent Samples Independent samples are those samples selected from two or more populations in such a way that the occurrence of values in one sample have no influence on the probability of the occurrence of values in the other sample(s).