1. Week 8 Fundamentals of Hypothesis Testing: One-Sample Tests
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 tested Example: 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 Is = 20
likely if μ = 50?
Suppose
the sample
If not likely,
REJECT
mean age
is 20: X = 20
Sample
Null Hypothesis

9. Sampling Distribution of
H0: μ = H1: μ ¹ 50
There are two cutoff values (critical values), defining the regions of rejection
/2
/2 X 50
Reject H0
Do not reject H0
Reject H0 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
Represents
critical value a a
H0: μ = 3 H1: μ ≠ 3 /2 /2
Rejection region is shaded
Two tailed test
H0: μ ≤ 3 H1: μ > 3 a
Upper tail test
H0: μ ≥ 3 H1: μ < 3 a
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 β

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

14. Outcomes and Probabilities
Possible Hypothesis Test Outcomes
Actual Situation
Decision
H0 True
H0 False
Key:
Outcome
(Probability)
Do Not
No error
( )
Type II Error
( β )
Reject a H
Reject
Type I Error
( )
No Error
( 1 - β ) H a

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 ( β )

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 <  , 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, α
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
 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 = 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
σ is unknown so this is a t test
5. Collect the data
Suppose the sample results are
n = 100, = s = 0.8
6. So the test statistic is:
The p value for n=100, =.05, t=-2 is .048

22. Hypothesis Testing Example
(continued)
7. Is the test statistic in the rejection region?
Reject H0 if p is < alpha; otherwise do not reject H0
The p-value .048 is < 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 < 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 < 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 > 3

25. Lower Tail Tests a H0: μ ≥ 3 H1: μ < 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 > 3
There is only one critical value, since the rejection area is in only one tail a
Do not reject H0
Reject H0 t tα 3
Critical value

27. Assumptions of the One-Sample t Test
The data is randomly selected
The population is normally distributed or the 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 Sample proportion in the success category is denoted by ps
(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 < 5 or
n(1-p) < 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 = H1: p ¹ .08
a = .05
n = 500, ps = .05
Critical Values: ± 1.96
p-value for is .0134
Decision:
Reject H0 at  = .05
Reject
Reject
.025
.025
There is sufficient evidence to reject the company’s claim of 8% response rate.
Conclusion: z
-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, α, 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 p–value approaches to hypothesis testing
Discussed type 1 and Type2 errors
Performed two tailed t test for the mean (σ unknown)
Performed Z test for the proportion
Discussed one-tail and two-tail tests
Addressed pitfalls and ethical issues