1. WELCOME BACK! 2017 128 days From: Tuesday, January 3, 2017
To: Thursday, May 11, 2017
128 days
Or 
4 months, 8 days 
11,059,200 seconds
184,320 minutes
3072 hours
18 weeks and 2 days
35.07% of 2017

2. Warm-Up: What makes a single die FAIR?
H0: The Prop. Of 1’s =2’s = … = 6’s OR The Die is Fair!
Ha: The Outcomes are not Uniformly distributed OR The Die is NOT Fair.

3. Two Proportions Z-Test
What is the statistical inference test you would use to determine if there is a significant difference in the true proportions of males with brown eyes vs. females with brown eyes?
Two Proportions Z-Test
Is there a Statistical Inference Test you would use if you needed to determine if there was a Significant Difference between Three or More Proportions?
YES!

4. (Degree of Freedom = df = n – 1).
Multiple Proportions
Chapter 26: Chi Square(d) or X2–Test
The P-Values/Probabilities for the X 2 – Test come from a family of Chi Square Distributions, which only take Positive Values and are all Skewed Right. A specific distribution is specified by a parameter called the Degree of freedom (df).
(Degree of Freedom = df = n – 1).

5. Calculating The X2 - Test Statistic:
Calculating The X2 - P-Value:
Or find the appropriate line on the X2 Table.
Find the P-Value for a Chi-Square Statistic = with df = 6.
P-Value = X2cdf (12.132, E99, 6) =

6. Ch. 26 - Multiple Proportions
There are THREE types of Chi-Square Tests:
1. The Chi-Square Test for Goodness of Fit.
2. The Chi-Square Test for Independence.
3. The Chi-Square Test for Homogeneity.

7. GOODNESS OF FIT The Chi-Square Test for
A test of whether the distribution of Counts in one categorical variable matches the distribution predicted (expected) by a model.
(Degree of Freedom = df = n – 1)
Where n = # of levels of the Category.
CONDITIONS
SRS
All Expected Counts greater than 5.

8. EXAMPLE: Is there one month of the year that stands out as having more births occurring as compared to the others? If births were distributed uniformly across the year, we would expect 1/12 of them to occur each month. To test the claim, birth data was randomly collected and compiled.
JAN.
FEB.
MAR
APR.
MAY
JUN.
JUL.
AUG.
SEP.
OCT.
NOV.
DEC.
OBS.
DATA 75 87 91 88 76 98 74 81 70 83
EXP.
DATA 82
H0: Births are uniformly distributed over the year.
Ha: Births are NOT uniformly distributed over the year. or
X2 Goodness of Fit Test
H0: Proportion of births in Jan = Feb.= Mar.= • • • = Dec.
Ha: Prop. of births are not all uniform. Not all pi’s equal.
P-Value =
X2cdf (9.5366, E99, 11) =
X2 =

9. 82 CONDITIONS SRS √ All Expected Counts are 5 or greater.
Since the P-Value is NOT less than α = we Fail to REJECT H0 . No evidence that Births do NOT occur uniformly through out the year.
CONDITIONS
SRS √
All Expected Counts are 5 or greater.
EXP.
DATA 82

10. Homework
Page 628: #3, 10

11. Homework: Page 628: #10

12. Homework
Page 628: #3-5, 9

13. EXAMPLE: The NY Civil Liberties Union feel that the NYC Police Dept is not hiring an ‘ethnic composition’ representing the city. NYC is 29.2% White, 28.3% Black, 31.5% Latino, 9.1% Asian, and 2% other. If the NYC Police Dept. is composed of the following, does the Union have a case?
White
Black
Latino
Asian
Other
OBS.
DATA
8560
7120
2762
1852
560
EXP.
DATA
6089.4
5901.7
6569
1897.7
417.08
H0: The Police Dept. represents the Population of NYC.
Ha: The Police Dept. Does NOT represents the Population of NYC.
X2 Goodness of Fit Test
P-Value =
X2cdf (3510.3, E99, 4)
= 0
X2 =

14. Since the P-Value is less than α = 0. 05 the data IS significant
Since the P-Value is less than α = the data IS significant . There is STRONG evidence to REJECT H0 . The hiring practice of the NYC police dept. does NOT represent the ethnic composition of NYC.
CONDITIONS
SRS X
All Expected Counts are 1 or greater. √
No more than 20% of the Expected Counts are less than 5. √
EXP.
DATA
6089.4
5901.7
6569
1897.7
417.08