Download Hypothesis Testing --- One Mean

Hypothesis Testing --- One Mean
A hypothesis is simply a statement that something is true.
Typically, there are two hypotheses in a hypothesis test:
the null, and
the alternative.
Null Hypothesis
The hypothesis to be tested is defined as a Null Hypothesis. We
use the symbol Ho to stand for null hypothesis.
Alternative Hypothesis
The Alternative Hypothesis is defined as a hypothesis that is
considered as an alternative to the null hypothesis. We use the
symbol Ha to stand for alternative hypothesis.
Important Considerations
1. Must contain an equality
2. Tests are always conducted assuming that the null is an
equality.
3. The most serious error would be the rejection of a true null.
Chapter 9: Hypothesis Testing: One Mean
Class Notes to accompany: Introductory Statistics, 9th Ed, By Neil A. Weiss
Prepared by: Nina Kajiji
Page -1-
Choosing & Specifying Hypothesis
The null hypothesis must always specify a single value when
testing population parameters.
The alternative hypothesis may be specified as not equal to,
greater than, or less than. That is:
Ho:
Ha:
=
Ho:
Ha:
=
<
Ho:
Ha:
=
>
0
0
--- Two-tailed test
0
0
--- Left-tailed test
0
0
--- Right-tailed test
Example:
A car manufacturer claims that a certain car gives a gas mileage of
26MPG. Consumer Agency doubts that the claim is true and
wants to test it. The hypothesis would be stated as a left-tailed
test.
The null hypothesis -- Ho : µ= 26
The alternate hypothesis -- Ha : µ < 26;
Chapter 9: Hypothesis Testing: One Mean
Class Notes to accompany: Introductory Statistics, 9th Ed, By Neil A. Weiss
Prepared by: Nina Kajiji
Page -2-
Pre-Steps To Testing The Hypothesis:
1.
2.
3.
4.
Take a random sample of observations – gas mileage achieved.
Compute the mean.
State the hypothesis
Test the hypothesis.
Motivation For Hypothesis Testing
Goal is to test if the population (sample) parameter (statistic) is
within a specified acceptance region. If it is then we accept the
null hypothesis and conclude that the alternative hypothesis is not
true.
On the other hand, if the parameter (statistic) value is outside the
acceptance region we reject the null hypothesis and conclude that
the alternative is true.
Some Key Terms
1. The test statistic used to conduct a hypothesis test is:
z
x
0
n
2. Rejection region or Critical region is a set of values for the test
statistic that leads to rejection of the null hypothesis.
For a two-tailed test the critical region is in both tails.
For a left-tailed test the critical region is in the left tail
For a right-tailed test the critical region is in the right tail.
Chapter 9: Hypothesis Testing: One Mean
Class Notes to accompany: Introductory Statistics, 9th Ed, By Neil A. Weiss
Prepared by: Nina Kajiji
Page -3-
3. Nonrejection region is a set of values for the test statistic that
lead to nonrejection of the null hypothesis.
4. Critical Values are the values of the test statistic that separate
the rejection and nonrejection regions.
5. Some Important Critical Values
z0.10 = 1.28
z0.05 = 1.645
z0.025 = 1.96
z0.01 = 2.33
z0.005 = 2.575
6. Significance level, , of a hypothesis test is defined to be the
probability of making a Type I error.
Example Cont:
Assume = 0.10; critical z = -1.28; n=30; x 25; σ = 1.4; and,
= 0.26
n
25 26
3.91  -3.91 < -1.28 thus, reject the null.
Then, z
0.26
Chapter 9: Hypothesis Testing: One Mean
Class Notes to accompany: Introductory Statistics, 9th Ed, By Neil A. Weiss
Prepared by: Nina Kajiji
Page -4-
Type I and Type II Errors
Decisions\Ho
Do not reject Ho
Reject Ho
True
Correct Decision
Type I Error
False
Type II Error
Correct Decision
Conclusion:
Type I error is rejecting the null hypothesis when it is in fact true.
Type II error is not rejecting the hypothesis when it is in fact false.
The smaller the Type I error probability, , of rejecting a true null
hypothesis, the larger the Type II error probability, , of not
rejecting a false null hypothesis; and vice versa.
Controlling Type I and Type II Errors
1. For a fixed alpha, an increase in sample size will cause a
decrease in beta.
2. For any fixed sample size, a decrease in alpha will cause an
increase in beta, and vice versa.
3. To decrease both alpha and beta, increase sample size.
Example:
A person is arrested and we have to decide based on some evidence
whether the person is innocent or guilty. Let, Ho = Innocent; and Ha =
Guilty. If we convict a person who is innocent we will commit a Type I
Error. If a guilty person goes free we have committed a Type II Error. If
one chooses a small we will have very few innocent people convicted;
but many guilty people might go free.
Chapter 9: Hypothesis Testing: One Mean
Class Notes to accompany: Introductory Statistics, 9th Ed, By Neil A. Weiss
Prepared by: Nina Kajiji
Page -5-
Hypothesis Tests For a Population Mean when Sigma is Known
Critical Value Approach
1. State the null and alternative hypothesis
2. Decide on the significance level,
3. Determine the critical values based on the test you are
conducting. That is:
a) for a two-tailed test CVs are z
/2
b) for a left-tailed test CV is -z
c) for a right-tailed test CV is z
4. Compute the value of the z test statistic.
5. If the value of the test statistic falls within the rejection region,
then reject Ho; otherwise do not reject Ho.
6. State the conclusion in words.
Possible Conclusion:
If the null hypothesis is rejected, we conclude that the
alternative hypothesis is probably true.
If the null hypothesis is not rejected, we conclude that the data
do not provide sufficient evidence to support the alternative
hypothesis.
See Example – 9.5
Chapter 9: Hypothesis Testing: One Mean
Class Notes to accompany: Introductory Statistics, 9th Ed, By Neil A. Weiss
Prepared by: Nina Kajiji
Page -6-
Example: Cellphone Bill
Is there enough evidence to suggest that the mean cellphone bill
has decreased from the base year?
Given: n=50; = 25; = 0.01
State Hypothesis:
Ho: = 47.37
Ha: < 47.37
Compute: z
41.08 47.37
25
50
0=47.37;
1.78
x
41.08
2.33
Do not reject the null. We do not have sufficient evidence to
conclude the mean bill has decreased from the base year.
Chapter 9: Hypothesis Testing: One Mean
Class Notes to accompany: Introductory Statistics, 9th Ed, By Neil A. Weiss
Prepared by: Nina Kajiji
Page -7-
P-Value Approach
The p-value of a hypothesis test is the probability of observing a
value of the test statistic that is at least as inconsistent with the
null hypothesis as the value of the test statistic actually observed.
The p-value is also frequently referred to as the observed
significance level or the probability value.
Using the P-Value to Assess the Evidence Against Ho
Common p-values Evidence against Ho
P > 0.10
Weak or none
Moderate
0.05 < P 0.10
Strong
0.01 < P 0.05
Very strong
P 0.01
Steps:
1. State the null & alternative hypotheses
2. Decide on the significance level,
3. Compute the value of the test statistic.
4. Determine the p-value for the test statistic. That is, the area in
the tail.
5. If the p-value is less than or equal to , reject Ho; otherwise, do
not reject Ho.
6. State the conclusion in words.
Chapter 9: Hypothesis Testing: One Mean
Class Notes to accompany: Introductory Statistics, 9th Ed, By Neil A. Weiss
Prepared by: Nina Kajiji
Page -8-
P-value for Cellphone Bill Example
-- smallest value of alpha to reject the null.
= P(z < -1.78) = 0.375 ; since 0.0375 > 0.01 we do not reject
the null.
Chapter 9: Hypothesis Testing: One Mean
Class Notes to accompany: Introductory Statistics, 9th Ed, By Neil A. Weiss
Prepared by: Nina Kajiji
Page -9-
Hypothesis Tests For One Population Mean when Sigma is
Unknown
1. State the null and alternative hypothesis
2. Decide on the significance level,
3. Determine the critical values based on the test you are
conducting with df = n-1. That is:
a) for a two-tailed test CVs are t
/2
b) for a left-tailed test CV is -t
c) for a right-tailed test CV is t
4. Compute the value of the test statistic. NOTE: you will need
to compute the standard deviation from the data.
5. If the value of the test statistic falls within the rejection region,
then reject Ho; otherwise do not reject Ho.
6. State the conclusion in words.
See Example: 9.16
Chapter 9: Hypothesis Testing: One Mean
Class Notes to accompany: Introductory Statistics, 9th Ed, By Neil A. Weiss
Prepared by: Nina Kajiji
Page -10-
Example
Bureau of Labor Statistics said that the average consumer unit
spent $1749 on apparel and services in 2002. Given that the mean
and standard deviation of 25 consumer units in the Northeast is
$1935.76 and $350.90 respectively, do the data provide sufficient
evidence at the 5% level to conclude that the 2002 mean
expenditures in the Northeast are significantly different from the
National average?
Given: n=25; s = 350.90; = 0.05
0=1749;
x 1935.76
State Hypothesis:
Ho: = 1749
Ha: <> 1749
Compute: t
1935.76 1749
2.661 2.064
350.90
25
Reject the null hypothesis. We have sufficient evidence to
conclude that consumer spending on apparel and service in the
Northeast is significantly different from the National average.
p-value = 2P(t > 2.661) = 2(0.005) = 0.01 —actually it is greater
than 0.01 but less than 0.02
Chapter 9: Hypothesis Testing: One Mean
Class Notes to accompany: Introductory Statistics, 9th Ed, By Neil A. Weiss
Prepared by: Nina Kajiji
Page -11-