Title: Week 8
1Statistics
- Week 8
- Fundamentals of Hypothesis Testing One-Sample
Tests
2Goals of this note
- After completing this noe, you should be able to
- Formulate null and alternative hypotheses for
applications involving a single population mean
or proportion - Formulate a decision rule for testing a
hypothesis - Know how to use the p-value approaches to test
the null hypothesis for both mean and proportion
problems - Know what Type I and Type II errors are
3What is a Hypothesis?
- A hypothesis is a claim
- (assumption) about a
- population parameter
- population mean
- population proportion
- The average number of TV sets in U.S. homes is
equal to three ( )
A marketing company claims that it receives 8
responses from its mailing. ( p.08 )
4The Null Hypothesis, H0
- States the assumption to be testedExample The
average number of TV sets in U.S. Homes is equal
to three ( ) - Is always about a population parameter,
not about a sample statistic
5The Null Hypothesis, H0
(continued)
- Begins with the assumption that the null
hypothesis is true - Similar to the notion of innocent until proven
guilty - Refers to the status quo
- Always contains , or ? sign
- May or may not be rejected
6The Alternative Hypothesis, H1
- Is the opposite of the null hypothesis
- e.g. The average number of TV sets in U.S. homes
is not equal to 3 ( H1 µ ? 3 ) - Challenges the status quo
- Never contains the , or ? sign
- Is generally the hypothesis that is believed (or
needs to be supported) by the researcher
7Hypothesis Testing
- We assume the null hypothesis is true
- If the null hypothesis is rejected we have proven
the alternate hypothesis - If the null hypothesis is not rejected we have
proven nothing as the sample size may have been
to small
8Hypothesis Testing Process
Claim the
population
mean age is 50.
(Null Hypothesis
Population
H0 µ 50 )
Now select a random sample
X
likely if µ 50?
20
Is
Suppose the sample
If not likely,
REJECT
mean age is 20 X 20
Sample
Null Hypothesis
9Sampling Distribution of
H0 µ 50 H1 µ ¹ 50
- There are two cutoff values (critical values),
defining the regions of rejection
?/2
?/2
X
50
Do not reject H0
Reject H0
Reject H0
0
20
Likely Sample Results
Lower critical value
Upper critical value
10Level of Significance, ?
- Defines the unlikely values of the sample
statistic if the null hypothesis is true - Defines rejection region of the sampling
distribution - Is designated by ? , (level of significance)
- Typical values are .01, .05, or .10
- Is the compliment of the confidence coefficient
- Is selected by the researcher before sampling
- Provides the critical value of the test
11Level of Significance and the Rejection Region
a
Represents critical value
Level of significance
a
a
/2
/2
H0 µ 3 H1 µ ? 3
Rejection region is shaded
Two tailed test
0
H0 µ 3 H1 µ gt 3
a
0
Upper tail test
H0 µ 3 H1 µ lt 3
a
0
Lower tail test
12Errors in Making Decisions
- Type I Error
- When a true null hypothesis is rejected
- The probability of a Type I Error is ?
- Called level of significance of the test
- Set by researcher in advance
- Type II Error
- Failure to reject a false null hypothesis
- The probability of a Type II Error is ß
13Example
Possible Jury Trial Outcomes
The Truth
Verdict
Guilty
Innocent
Innocent
No error
Type II Error
Guilty
Type I Error
No Error
14Outcomes and Probabilities
Possible Hypothesis Test Outcomes
Actual Situation
Decision
H0 False
H0 True
Key Outcome (Probability)
Do Not
No error (1 - )
Type II Error ( ß )
Reject
a
H
0
Reject
Type I Error ( )
No Error ( 1 - ß )
H
a
0
15Type I II Error Relationship
- Type I and Type II errors can not happen at
- the same time
- Type I error can only occur if H0 is true
- Type II error can only occur if H0 is false
- If Type I error probability ( ? ) , then
- Type II error probability ( ß )
16p-Value Approach to Testing
- p-value Probability of obtaining a test
statistic more extreme ( or ? ) than the
observed sample value given H0 is true - Also called observed level of significance
17p-Value Approach to Testing
(continued)
- Convert Sample Statistic (e.g. ) to Test
Statistic (e.g. t statistic ) - Obtain the p-value from a table or computer
- Compare the p-value with ?
- If p-value lt ? , reject H0
- If p-value ? ? , do not reject H0
X
188 Steps in Hypothesis Testing
- 1. State the null hypothesis, H0
- State the alternative hypotheses, H1
- 2. Choose the level of significance, a
- 3. Choose the sample size, n
- 4. Determine the appropriate test statistic to
use - 5. Collect the data
- 6. Compute the p-value for the test statistic
from the sample result - 7. Make the statistical decision Reject H0 if
the p-value is less than alpha - 8. Express the conclusion in the context of the
problem
19Hypothesis Tests for the Mean
Hypothesis Tests for ?
? Known
? Unknown
20Hypothesis Testing Example
Test the claim that the true mean of TV sets in
U.S. homes is equal to 3.
- 1. State the appropriate null and alternative
- hypotheses
- H0 µ 3 H1 µ ? 3 (This is a two
tailed test) - 2. Specify the desired level of significance
- Suppose that ? .05 is chosen for this test
- 3. Choose a sample size
- Suppose a sample of size n 100 is selected
21Hypothesis Testing Example
(continued)
- 4. Determine the appropriate Test
- s is unknown so this is a t test
- 5. Collect the data
- Suppose the sample results are
- n 100, 2.84 s 0.8
- 6. So the test statistic is
- The p value for n100, ?.05, t-2 is .048
22Hypothesis Testing Example
(continued)
- 7. Is the test statistic in the rejection region?
Reject H0 if p is lt alpha otherwise do not
reject H0
The p-value .048 is lt alpha .05, we reject the
null hypothesis
23Hypothesis Testing Example
(continued)
- 8. Express the conclusion in the context of the
problem - Since The p-value .048 is lt alpha .05,
we have rejected the null hypothesis
Thereby proving the alternate hypothesis - Conclusion There is sufficient evidence
that the mean number of TVs in U.S. homes is
not equal to 3
If we had failed to reject the null hypothesis
the conclusion would have been There is not
sufficient evidence to reject the claim that the
mean number of TVs in U.S. home is 3
24One Tail Tests
- In many cases, the alternative hypothesis focuses
on a particular direction
This is a lower tail test since the alternative
hypothesis is focused on the lower tail below the
mean of 3
H0 µ 3 H1 µ lt 3
This is an upper tail test since the alternative
hypothesis is focused on the upper tail above the
mean of 3
H0 µ 3 H1 µ gt 3
25Lower Tail Tests
H0 µ 3 H1 µ lt 3
- There is only one critical value, since the
rejection area is in only one tail
a
Reject H0
Do not reject H0
-t
3
Critical value
26Upper Tail Tests
H0 µ 3 H1 µ gt 3
- There is only one critical value, since the
rejection area is in only one tail
a
Reject H0
Do not reject H0
t
ta
3
Critical value
27Assumptions of the One-Sample t Test
- The data is randomly selected
- The population is normally distributed orthe
sample size is over 30 and the population is not
highly skewed
28Hypothesis Tests for Proportions
- Involves categorical values
- Two possible outcomes
- Success (possesses a certain characteristic)
- Failure (does not possesses that
characteristic) - Fraction or proportion of the population in the
success category is denoted by p
29Proportions
(continued)
- Sample proportion in the success category is
denoted by ps -
- When both np and n(1-p) are at least 5, ps
can be approximated by a normal distribution with
mean and standard deviation -
30Hypothesis Tests for Proportions
- The sampling distribution of ps is
approximately normal, so the test statistic is a
Z value
Hypothesis Tests for p
np ? 5 and n(1-p) ? 5
np lt 5 or n(1-p) lt 5
Not discussed in this chapter
31Example Z Test for Proportion
- A marketing company claims that it receives 8
responses from its mailing. To test this claim,
a random sample of 500 were surveyed with 25
responses. Test at the ? .05 significance
level.
Check n p (500)(.08) 40 n(1-p)
(500)(.92) 460
?
32Z Test for Proportion Solution
Test Statistic
H0 p .08 H1 p ¹ .08
Critical Values 1.96
p-value for -2.47 is .0134 Decision Reject H0 at
? .05
Reject
Reject
.025
.025
There is sufficient evidence to reject the
companys claim of 8 response rate.
Conclusion
z
0
1.96
-1.96
-2.47
33Potential Pitfalls and Ethical Considerations
- Use randomly collected data to reduce selection
biases - Do not use human subjects without informed
consent - Choose the level of significance, a, before data
collection - Do not employ data snooping to choose between
one-tail and two-tail test, or to determine the
level of significance - Do not practice data cleansing to hide
observations that do not support a stated
hypothesis - Report all pertinent findings
34Summary
- Addressed hypothesis testing methodology
- Discussed critical value and pvalue approaches
to hypothesis testing - Discussed type 1 and Type2 errors
- Performed two tailed t test for the mean (s
unknown) - Performed Z test for the proportion
- Discussed one-tail and two-tail tests
- Addressed pitfalls and ethical issues