Week 8

1
Statistics

• Week 8
• Fundamentals of Hypothesis Testing One-Sample
  Tests

2
Goals 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

3
What 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 )
4
The 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

5
The 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

6
The 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

7
Hypothesis 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

8
Hypothesis 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
9
Sampling 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
10
Level 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

11
Level 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
12
Errors 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

13
Example
Possible Jury Trial Outcomes
The Truth
Verdict
Guilty
Innocent
Innocent
No error
Type II Error
Guilty
Type I Error
No Error
14
Outcomes 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
15
Type 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 )

16
p-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

17
p-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
18
8 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

19
Hypothesis Tests for the Mean
Hypothesis Tests for ?
? Known
? Unknown
20
Hypothesis 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

21
Hypothesis 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

22
Hypothesis 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
23
Hypothesis 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
24
One 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
25
Lower 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
26
Upper 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
27
Assumptions 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

28
Hypothesis 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

29
Proportions
(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

30
Hypothesis 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
31
Example 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
?
32
Z Test for Proportion Solution
Test Statistic
H0 p .08 H1 p ¹ .08

• a .05
• n 500, ps .05

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
33
Potential 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

34
Summary

• 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