X 2 Test of Independence

1. WARM-UP: To Examine whether the grades on the AP Statistics Fall Exam were independent of what class period you are in, a SRS of 70 students were selected. X2 Test of Independence H0: Test Grades and Class Period are independent. Ha: The Grade you earn on the Test is associated to what class period you are in. P-Value = X2cdf (8.75, E99, 6) = 0.1881 X2 = 8.751 Since the P-Value is NOT less than α = 0.05 you will Fail to reject H0. There is no evidence to conclude that Test Grades and Class Period are related. • SRS - Stated√ • All Expected Counts are 5 or greater. X PWC

2. Homework check

3. The Chi-Square Test for Homogeneity A test comparing the distribution of counts for TWO or MOREPopulations on the ONE categorical variable. -GOF tests only one Population on only ONE categorical variable. -Homogeneity represents Multiple GOF tests. df = (#Rows – 1)x (#Cols. – 1) H0: The distribution of the one variable is equivalent among the populations Ha: The distribution of the one variable is NOT equivalent among the populations

4. NOTE: The Chi-Square Tests for Homogeneity and for Independence are performed exactly the same way! P-Value = X2cdf (X2, E99, df)

5. EXAMPLE:Is the Distribution of colors in a package of PLAIN M&M’s statistically equivalent to the Distribution of colors in a package of PEANUT M&M’s? A random package of plain and peanut M&M’s are selected and analyzed. X2 Test of Homogeneity H0: The Distribution of colors in the Plain Packet of M&M’s is equivalent to that of the Peanut M&M’s. Ha: The Distribution of colors in the Plain Packet of M&M’s is NOT equivalent to that of the Peanut M&M’s. P-Value = X2cdf (4.967, E99, 5) = 0.4200 X2 = 4.967

6. EXAMPLE:Is the Distribution of colors in a package of PLAIN M&M’s statistically equivalent to the Distribution of colors in a package of PEANUT M&M’s? A random package of plain and peanut M&M’s are selected and analyzed. X2 Test of Homogeneity H0: The Distribution of colors in the Plain Packet of M&M’s is equivalent to that of the Peanut M&M’s. Ha: The Distribution of colors in the Plain Packet of M&M’s is NOT equivalent to that of the Peanut M&M’s. P-Value = X2cdf (4.967, E99, 3) = 0.4200 X2 = 4.967 • CONDITIONS • SRS - Stated √ • All Expected Counts are 5 or greater.X Since the P-Value is NOT less than α = 0.05 fail to reject H0. No evidence that the Distributions are NOT equivalent. Although the results are uncertain.

7. How well do YOU feel JJ Pearce High School is preparing you for the Real World (College and/or job Post High School)? Is the Distribution of School Readiness attitudes statistically equivalent among each gender?

8. X2 Test of Homogeneity How well do YOU feel JJ Pearce High School is preparing you for the Real World (College and/or job Post High School)? H0: The Distribution of Readiness Attitudes are equivalent among each gender. Ha: The Distribution of Readiness Attitudes are NOT equivalent among each gender. P-Value = ________ X2 = ______

9. WARM-UP:Examine whether the distribution of grades is equivalent for each period of AP Statistics. X2 Test of HOMOGENEITY H0: The distribution of A,B,C, and F’s in all three periods is spread equivalently. Ha: The distribution of A,B,C, and F’s in all three periods is NOT spread equivalently. X2 = 8.75 P-Value = X2cdf (8.75, E99, 6) = .1881 Since the P-Value is NOT less than α = 0.05 we fail to reject H0. There is no evidence to conclude that Test Grades are NOT distributed equally among the 3 classes. • SRS - StatedX • All Expected Counts are 1 or greater.√ • No more than 20% of the Expected Counts are less than 5. X

10. Homework: Page 630: #15 omit h, 16, 17, & 20

12. Homework: Page 630: 17, 20, 21

13. An SRS of 120 voters from AR and an SRS of 115 voters from TX was taken to determine whether there was a significant difference in how people, as of that moment, would vote with regards to Clinton. X2 Test of Homogeneity H0: The Distribution of how people would vote today in the State of Arkansas is equal to that of Texas. Ha: The Distribution of how people would vote today in the State of Arkansas is NOT equal to that of Texas. P-Value = X2cdf (11.277, E99, 3) = 0.0103 X2 = 11.277 Since the P-Value is less than α = 0.05 the data IS significant . REJECT H0 . Support is different between AR and TX. • SRS – stated • All Expected Counts are 5 or greater.

14. #18 Medical researchers followed an SRS of 6272 Swedish men for 30 years to see if there was an association between the amount of fish in their diet and Prostate Cancer. Is there any evidence of such an association? H0: There is NO relationship between fish consumption and the development of Prostate Cancer. Ha: There is relationship between fish consumption and the development of Prostate Cancer. X2 Test of Independence P-Value = X2cdf (3.677, E99, 3) = 0.2985 X2 = 3.677

15. WARM – UP Medical researchers followed 6272 Swedish men for 30 years to see if there was an association between the amount of fish in their diet and Prostate Cancer. Is there any evidence of such an association? H0: There is NO relationship between fish consumption and the development of Prostate Cancer. Ha: There is relationship between fish consumption and the development of Prostate Cancer. X2 Test of Independence P-Value = X2cdf (3.677, E99, 3) = 0.2985 X2 = 3.677 • CONDITIONS • SRS - Stated √ • All Expected Counts are 1 or greater.√ • No more than 20% of the Expected Counts are less than 5. √ Since the P-Value is NOT less than α = 0.05 there is NO evidence to reject H0. There is NO relationship between fish consumption and Prostate Cancer.

16. WARM – UP Does ones regional location have an affect on their Political affiliation? To begin to investigate this situation data from 177 voters was analyzed. % of Dem. 0 50 100 N NW SE Regional Location Political Affiliation 33.93 0.591 32.07 Location 25.71 0.700 24.29 a.) Find the Proportion of Democrats in each region. b.) Make a Bar Chart for the Prop. c.) Find the Expected Values for each cell. 31.36 0.279 29.64