Chapter 8 Hypothesis Testing - Annville-Cleona School District...Chapter 8 Hypothesis Testing...
Transcript of Chapter 8 Hypothesis Testing - Annville-Cleona School District...Chapter 8 Hypothesis Testing...
Chapter 8 Hypothesis
Testing
Section 8.1 Introduction to Statistical Tests
It is used to make decisions concerning the
value of a parameter.
Hypothesis testing
It is used to make decisions concerning the value of a
parameter.
Hypothesis testing
Null Hypothesis: H0
A working hypothesis about the population
parameter in question.
The value specified in the null hypothesis is often:
Øa historical value Øa claim Øa production specification
Alternate Hypothesis: H1
any hypothesis that differs from the null
hypothesis
An alternate hypothesis is constructed in such a way that it is the one to be accepted when the
null hypothesis must be rejected.
A manufacturer claims that their light bulbs burn for an average of 1000 hours. We have reason to believe that the bulbs do not last that long. Determine the null and alternate hypotheses.
A manufacturer claims that their light bulbs burn for an
average of 1000 hours. ...
The null hypothesis (the claim) is that the true average life is
1000 hours.
H0: µ = 1000
… A manufacturer claims that their light bulbs burn for an
average of 1000 hours. We have reason to believe that the bulbs
do not last that long. ...
If we reject the manufacturer’s claim, we must accept the alternate
hypothesis that the light bulbs do not last as long as 1000 hours.
H1: µ < 1000
Type I Error
Rejecting a null hypothesis which
is in fact true
Type II Error
Not rejecting a null hypothesis which
is in fact false
Options in Hypothesis Testing
H0 is
Our Choices:
Do Not Reject Reject
True
False
Type I Error
Correct DecisionType II Error
Correct Decision
Level of Significance, Alpha
The probability with which we are
willing to risk a type I error
α( )
Type II Error➡β = beta = probability of a type II
error (failing to reject a false hypothesis)
➡Α small α is normally is associated with a (relatively) large β, and vice-versa.
➡Choices should be made according to which error is more serious.
Power of the Test 1 – Beta
➡The probability of rejecting H0 when it is in fact false = 1 – β.
➡The power of the test increases as the level of significance (α) increases.
➡Using a larger value of alpha increases the power of the test but also increases the probability of rejecting a true hypothesis.
Probabilities Associated with a Hypothesis Test
Do Not Reject H0
Reject H0
H0 is True
H0 is False
Type I Error with probability α
Correct Decision with probability
of 1 - α
Type II Error with probability of β
Correct Decision with probability
of 1 - β
Reject or ...➡ When the sample evidence is not
strong enough to justify rejection of the null hypothesis, we fail to reject the null hypothesis.
➡ Use of the term “accept the null hypothesis” should be avoided.
➡ When the null hypothesis cannot be rejected, a confidence interval is frequently used to give a range of possible values for the parameter.
Fail to Reject H0
There is not enough evidence to reject H0. The null hypothesis is retained but not
proved.
Reject H0 There is enough
evidence to reject H0. Choose the alternate hypothesis with the
understanding that it has NOT been proven.
A fast food restaurant indicated that the average age of its job applicants is fifteen years. We suspect that the true age is lower than 15.
We wish to test the claim with a level of significance of α = 0.01.
… average age of its job applicants is fifteen years. We suspect that the true
age is lower than 15.H0: µ = 15 H1: µ < 15
Describe Type I and Type II errors.
H0: µ = 15 H1: µ < 15 α = 0.01
A type I error would occur if we rejected the claim that the mean age was 15, when in fact the mean age was 15 (or higher). The probability of committing such an error is as much as 1%.
H0: µ = 15 H1: µ < 15 α= 0.01
A type II error would occur if we failed to reject the claim that the mean age was 15, when in fact the mean age was lower than 15. The probability of committing such an error is called beta.
Types of TestsüWhen the alternate hypothesis
contains the “not equal to” symbol ( ≠ ), perform a two-tailed test.
ü When the alternate hypothesis contains the “greater than” symbol ( > ), perform a right-tailed test.
ü When the alternate hypothesis contains the “less than” symbol ( < ), perform a left-tailed test.
Two-Tailed Test
H0: µ = k
H1: µ ≠ k
Two-Tailed Test
If test statistic is in either tail - the critical region - of the distribution,
we REJECT the Null Hypothesis.
H0: µ = k H1: µ ≠ k
If test statistic is at or near the claimed
mean, we DO NOT REJECT
the Null Hypothesis
–z 0 z
Right-Tailed Test
H0: µ = k
H1: µ > k
Right-Tailed Test
If test statistic is in the right tail - the critical region - of the
distribution, weREJECT the Null Hypothesis.
H0: µ = k H1: µ > k
If test statistic is at, near, or
below the claimed
mean, we DO NOT REJECT
the Null Hypothesis
0 z
Left-Tailed Test
H0: µ = k
H1: µ < k
Left-Tailed Test
If test statistic is in the left tail - the critical region - of the distribution,
weREJECT the Null Hypothesis.
H0: µ = k H1: µ < k
If test statistic is at, near, or
above the claimed
mean, we DO NOT REJECT
the Null Hypothesis
-z 0
END OFSECTION
8.1