1. Lecture #8 Thursday, September 15, 2016 Textbook: Section 4.4
Statistics 200
Lecture #8 Thursday, September 15, 2016
Textbook: Section 4.4
Objectives (for two categorical variables and their relationship):
• Understand Simpson’s paradox: What causes it, why it’s surprising
• Formulate null and alternative hypotheses in a testing scenario
• Calculate chi-square statistic for a 2x2 table
• Interpret p-value derived from a chi-square statistic; make a decision based on this interpretation
• Contrast statistical significance and practical significance
• Understand how sample size affects statistical significance
• Also: Discuss Exam #1

2. On Monday we introduced a LOT of terms:
yes no
Total
Group1
Group2
2×2
Table
One Group:
from table
Individual Risk:
risk for
one group
Odds: compares two possible outcomes
within one group
Comparing
Two Groups:
(single number)
Relative Risk:
(ratio)
Increased Risk:
(percent)
yes no
Total
Group1
Group2

3. Todays objectives: Introduce hypothesis testing!
Assess the statistical significance of a two-way table using the chi-square test.

4. Statistical Significance
A statistically significant relationship or difference is one that is large enough to be unlikely to have occurred in the sample if there was no relationship or difference in the population.

5. Statistical significance: Hypotheses
We determine statistical significance by performing something called a hypothesis test.
For hypothesis testing, we first create two hypotheses, known as the null and alternative hypotheses.
State null and alternative hypotheses before looking at data.

6. Null Hypothesis: H0 Nothing population No No sample null
Starting Position:
__________ is happening
Nothing
Null: With (2 × 2 Tables):
In the ______________ there is:
____ relationship between the two variables
____ difference in the two groups
population No No
Statistical Hypotheses:
never include statements about ________
sample
null
Never trying to prove the ______ is true

7. Alternative Hypothesis: Ha
something
Challenging Position: ____________ is happening
data
Goal: hope the ________ supports
Alternative: With (2 × 2 Tables):
In the population, there is:
_____ relationship between the two variables
_____ difference in the two groups a a

8. Selfie Example: State Hypotheses
Yes No
Female 90
110
Male 30 70
Research Question:
Does the data suggest that there is a relationship between: (sex) and (whether or not a Stat 200 student likes to take “selfies”)?
When considering these two variables, in the _____________:
Null (H0): there is _____ relationship
(i.e. there is no difference in two groups when comparing “sex”)
Alternative (Ha): there is ____ relationship
(i.e. there is a difference in the two groups when comparing “sex”)
population no a

9. We quantify evidence for the alternative using…
Chi-Square statistic
Next step: Find chi-square statistic using sample data
Used to determine whether there is a significant relationship between two categorical variables.
Calculate using sample data

10. Chi-Square Statistic:
actual (observed) counts
chi-square statistic
quantifies the amount
of difference between
Yes No
Female 90
110
Male 30 70
expected counts
Expected Counts:
are hypothetical counts
are calculated using a _______________
based on the statement found in the ________ hypothesis
take the position of _______ relationship ( ____ difference)
formula
null no no

11. Statistical significance: Chi-Square
This statistic measures the difference in the observed counts and the expected counts in a contingency table.
When we talk about expected counts, we’re talking about the counts that we would expect to see if there is no relationship between the variables (the null hypothesis is true).

12. Calculating expected counts:
Yes No
Row Total
Female
200
Male
100
Column Total
120
180
300 = n
Expected count =
actual
Don’t need: ___________ counts

13. Expected count =
Yes No
Row Total
Female
200
Male
100
Column Total
120
180
300 = n
200 x 120
300
= 80
200 x 180
300
= 120
100 x 120
300
= 40
100 x 120
300
= 60

14. Expected counts in Minitab from raw data: Stat> Tables > Cross tabulation and Chi-Square
Explanatory Variable
Response Variable
actual data
expected counts

15. Minitab Output Actual count Expected count Rows: Sex Columns: Selfie
yes no All
female
male
All
Cell Contents: Count
Expected count
Actual count
Expected count
Difference between ‘actual’ and ‘expected’ is 10. This is also called the ‘residual’.

16. Use counts to calculate Chi-Square
The Chi-square statistic lets us use a number to describe how different the actual counts are from the expected counts.
Each cell in the table contributes:
Finally, we add all the contributions together to get the chi-square statistic.

17. Example Calculation: Chi-Square Statistic
Chi-Square statistic Formula:
LEAST
MOST = =
dEach cell contributes to Chi-square. We can tell which one contributes most and least

18. Chi-Square Statistic infinity larger
single number: combines information from all four cells
possible values: ___ to _______
If the null is true, we know how this value should behave. We can use this knowledge to learn how unusual our sample would be under the null.
the ______________ the value of chi-square, the more unusual our sample would be under the null, and the more evidence we have for our alternative hypothesis.
infinity
larger

19. Chi-square to p-value P-value
How large of a chi-square statistic do we need to declare significance?
Convert to a different question:
What is the likelihood that the chi-square statistics could be this large or larger if there is actually no relationship in the population?
We can answer this question using a
P-value

20. P-value Definition (Interpretation)
Box definition:
page 126
textbook
or any value
more ________
in the direction
stated in _________
when assuming the
_____
hypothesis
is true.
The p-value is the likelihood
of getting our
result
extreme
null
alternative

21. Use Minitab to calculate P-value
Example 1: Chi-Sq = 6.25
Use Minitab to calculate P-value
p-value = 0.012
p-value interpretation:
The likelihood of getting our chi-square statistic of _____
or any value more extreme (_______________)
when assuming there is ____ relationship in the population
is _______.
6.25
larger no
0.012

22. When is a p-value small enough to declare statistical significance?
The typical rule is to declare significance when the p-value is less than 0.05.
Common Language
Statistical
Statement
Guideline to make decision:
Result is:
“statistically significant”
unlikely due to random chance
can reject H0 in favor of Ha
p-value < .05
“ not statistically significant”
possibly due to random chance
can not reject H0 in favor of Ha
p-value > .05

23. Example Conclusion: Statistical Significance is Found?
p-value = (from output)
<
Our Conclusion: Since our p-value of
We can __________ H0 in favor of Ha
In the population: we ________ claim that there is:
_____ statistically significant relationship between sex and
whether or not a stat 200 student likes to take “selfies”
There is _____ difference between the two groups
We can rule out ___________ chance as an explanation for our result
reject
can a a
random

24. Summary :Hypothesis Tests
State in: hypotheses (null and alternative)
Complete the necessary calculations: expected counts, chi-square statistic, p-value
A one-sentence statement:
what does the p-value mean?
Compare: the result to an established guideline (.05) and reject or fail to reject H0

25. What if the p-value >.05?
Then the relationship between the two variables is not statistically significant
When you are interpreting this, you should say:
“We fail to reject the null hypothesis” or
“The p-value is too large to declare that there is a statistically significant relationship.”
We never ‘accept’ H0, we ‘fail to reject’ H0

26. What influences statistical significance?
The relationship between two variables can have its significance effected by two factors:
The strength of the observed relationship. How different are the row percents?
The number of people that were studied (sample size).

27. Example: Are women more likely to have dogs?
Has Dog
No Dog
Total
Female 89
56.7% 68
43.3%
157
Male 66
50.8% 64
49.2%
130
155
132
287
Your class data

28. Example: Are women more likely to have dogs?
Has Dog
No Dog
Total
Female 89
56.7% 68
43.3%
157
Male 66
50.8% 64
49.2%
130
155
132
287
Is there evidence here for a relationship between a person’s gender and whether or not they have a dog?

29. Example: Are women more likely to have dogs?
Is there evidence here for a relationship between a person’s gender and whether or not they have a dog?
Formulate and test statistical hypotheses
H0: There is no relationship in the population between gender and dog ownership.
Ha: There is a relationship in the population between gender and dog ownership.

30. Example: Are women more likely to have dogs?
Is there evidence here for a relationship between a person’s gender and whether or not they have a dog?
Formulate and test statistical hypotheses
Calculate the chi-square statistic and p-value
Chi-square statistic: 0.779
P-value: 0.378

31. Example: Are women more likely to have dogs?
Is there evidence here for a relationship between a person’s gender and whether or not they have a dog?
Formulate and test statistical hypotheses
Calculate the chi-square statistic and p-value
Interpretation of p-value:
The likelihood of getting a chi-square statistic of or larger is 0.378, assuming that there is truly no relationship in the population.

32. Example: Are women more likely to have dogs?
Is there evidence here for a relationship between a person’s gender and whether or not they have a dog?
Formulate and test statistical hypotheses
Calculate the chi-square statistic and p-value
Make a decision.
Based on the p-value of we fail to reject the null hypothesis. We cannot conclude that there is a relationship between gender and dog ownership in the population.

33. Now, what if we had a larger sample with the same row percentages?
Has Dog
No Dog
Total
Female 89
56.7% 68
43.3%
157
Male 66
50.8% 64
49.2%
130
155
132
287
P-value:
0.378
Has Dog
No Dog
Total
Female
890
56.7%
680
43.3%
1570
Male
660
50.8%
640
49.2%
1300
1550
1320
2870
P-value:
0.0053

34. Moral of the example: Sample P-value size
If the row percentages stay the same but the sample size is increased, then we have more evidence against the null hypothesis. This results in a smaller p-value.
Sample
size
P-value

35. Another issue: Practical vs. statistical significance
Remember from one of the morals of statistics that statistical significance does not always mean there is practical significance.
For example
If drug A is two times as likely to cure a cold as drug B, there will probably be statistical significance.
However, if drug B only works 0.001% of the time, and drug A works 0.002% of the time, the difference isn’t practically significant.

36. Review: If you understood today’s lecture, you should be able to solve
4.41, 4.43, 4.45, 4.47, 4.51, 4.53, 4.55abd, 4.61
Recall objectives:
• Formulate null and alternative hypotheses in a testing scenario
• Calculate chi-square statistic for a 2x2 table
• Interpret p-value derived from a chi-square statistic; make a decision based on this interpretation
• Contrast statistical significance and practical significance
• Understand how sample size affects statistical significance