1. Introduction to Inductive Statistics
Hypothesis Testing
Introduction to Inductive Statistics

2. Background Website

3. Terms Descriptive Statistics: what we’ve done so far…
Inductive Statistics: making decisions on the basis of statistical evidence
Hypothesis: the relationship or proposition you wish to test, stated to affirm the relationship or proposition
Null Hypothesis: the negative form of the proposition to be tested

4. Some propositions…
Statistics is making decisions on the basis of incomplete or imperfect information.
A hypothesis or proposition can be refuted by one observation but not proved by many.
We thus proceed by determining the likelihood that the null hypothesis can be rejected.

5. The Dilemma…
There is always the possibility of making the wrong decision:
Rejecting a true hypothesis: Type 1 Error
Failing to reject a false hypothesis: Type 2 Error

6. An Example of the Issue: A Jury’s Decision Making
Justice System - Trial
Defendant Innocent
Defendant Guilty
Reject Presumption of Innocence (Guilty Verdict)
Type I Error
Correct
Fail to Reject Presumption of Innocence (Not Guilty Verdict)
Type II Error
Truth: Unknown to the Jury
Jury’s Decision on the basis of evidence

7. An Example of the Issue: A Jury’s Decision Making
Justice System - Trial
Defendant Innocent
Defendant Guilty
Reject Presumption of Innocence (Guilty Verdict)
Type I Error:
False Positive Error
Correct
Fail to Reject Presumption of Innocence (Not Guilty Verdict)
Type II Error:
False Negative Error
Truth: Unknown to the Jury
Jury’s Decision on the basis of evidence

8. The Statistical Decision Making Framework
              Statistics - Hypothesis Test
Null Hypoth True
Null Hypoth False
Reject Null Hypothesis
Type I Error
Correct
Fail to Reject Null Hypothesis
Type II Error
Unknown Truth
Researcher’s Decision

9. The Statistical Decision Making Framework
              Statistics - Hypothesis Test
Null Hypoth True
Null Hypoth False
Reject Null Hypothesis
Type I Error:
False Positive Error
Correct
Fail to Reject Null Hypothesis
Type II Error:
False Negative Error
Unknown Truth
Researcher’s Decision

10. The Jury and the Researcher Compared
Justice System - Trial
Defendant Innocent
Defendant Guilty
Reject Presumption of Innocence (Guilty Verdict)
Type I Error
Correct
Fail to Reject Presumption of Innocence (Not Guilty Verdict)
Type II Error
              Statistics - Hypothesis Test
Null Hypoth True
Null Hypoth False
Reject Null Hypothesis
Type I Error
Correct
Fail to Reject Null Hypothesis
Type II Error

11. Steps for Making a Decision
Specify a hypothesis and the null hypothesis
Specify a level of probability which you will use to decide whether to reject the null hypothesis.
Specify the test statistic and the sampling distribution you will use to make a decision.
Calculate the statistics and compare to the theoretical probability distribution, for example, the t distribution.
Interpret the results.

12. General Form of the Test for a Mean
Z tests:
(Sample mean – population mean) / SE of sample mean, or
(Sample mean – population mean)/ (s / √n)
T tests:
(Sample mean – population mean)/ (s / √n-1)

13. General Form of a T Test t = sample estimate – null hypothesis/ SE
Which simplifies to:
t = sample estimate/SE
When the null hypothesis is that the sample statistic is 0.

14. T Distribution

15. Example 1
Hypothesis: There is a difference in the average number of persons per household in the 18th and the 14th wards.
Null Hypothesis: There is no difference in the average number of persons per household in the 18th and the 14th wards, or more specifically, any difference we measure is a matter of the particular sample we have.

16. Example, cont.
Level of probability: 95% confidence level, so that only 1 in 20 times would the results be different.
Test statistic: Means and a T-Test of the difference of two groups.
t = (mean1 – mean2)/ (SE of the difference of mean1-mean2)
Calculate the statistics….

17. Results Two-sample t test on PERSONS grouped by WARD Group N Mean SD
Separate Variance t = df = Prob =
Difference in Means = % CI = to
Pooled Variance t = df = Prob =
Difference in Means = % CI = to

18. Results, Graphically Displayed

19. Interpret the Results Let’s look at the t distribution again.
We can reject the null hypothesis that the two means are the same in the underlying population (the unknown truth).
We say that there is a statistically significant difference between the average number of persons in the two wards.

20. Example 2
Hypothesis: There is a difference in the average number of persons per household in the 18th and the 20th wards.
Null Hypothesis: There is no difference in the average number of persons per household in the 18th and the 20th wards, or more specifically, the difference is a matter of the particular sample we have.

21. Results… TEST PERSONS * WARD
Data for the following results were selected according to:
(WARD<> 14) AND (ward<> 22)
Two-sample t test on PERSONS grouped by WARD
Group N Mean SD
Separate Variance t = df = Prob =
Difference in Means = % CI = to
Pooled Variance t = df = Prob =
Difference in Means = % CI = to

22. Results, Graphically Displayed…

23. Interpret the Results
We cannot reject the null hypothesis that the two means are the same in the underlying population (the unknown truth).
We say that there is not a statistically significant difference between the average number of persons in the two wards.

24. An example of the issue: A jury’s decision making
Justice System - Trial
Defendant Innocent
Defendant Guilty
Reject Presumption of Innocence (Guilty Verdict)
Type I Error
Correct
Fail to Reject Presumption of Innocence (Not Guilty Verdict)
Type II Error