X2 = 8.751 X2 Test of Independence P-Value = 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. A B C F 1st Period 5 2 4 4rd Period 12 1 6th Period 16 8 3 7.5 3.4 3.0 2.1 10.8 4.9 4.3 14.6 6.6 5.8 4.0 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

Homework check

The Chi-Square Test for Homogeneity A test comparing the distribution of counts for TWO or MORE Populations 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

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

X2 = 4.967 X2 Test of Homogeneity P-Value = 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. 3.39 4.06 2.71 6.10 0.68 1.61 1.94 1.29 2.90 0.32 Brown Blue Orange Green Red Yellow PLAIN 4 3 2 7 5 PEANUT 1 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

X2 = 4.967 X2 Test of Homogeneity P-Value = 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. 3.39 4.06 2.71 6.10 0.68 1.61 1.94 1.29 2.90 0.32 Brown Blue Orange Green Red Yellow PLAIN 4 3 2 7 5 PEANUT 1 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.

Not sufficiently prepared Sufficiently prepared How well do YOU feel JJ Pearce High School is preparing you for the Real World (College and/or job Post High School)? Not at all prepared A Not sufficiently prepared B Sufficiently prepared C Very well prepared D Male Female Is the Distribution of School Readiness attitudes statistically equivalent among each gender?

Not sufficiently prepared Sufficiently prepared 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)? Not at all prepared A Not sufficiently prepared B Sufficiently prepared C Very well prepared D Male Female H0: The Distribution of Readiness Attitudes are equivalent among each gender. Ha: The Distribution of Readiness Attitudes are NOT equivalent among each gender. X2 = ______ P-Value = ________

X2 = 8.75 X2 Test of HOMOGENEITY WARM-UP: Examine whether the distribution of grades is equivalent for each period of AP Statistics. A B C F 1st Period 5 2 4 4rd Period 12 1 6th Period 16 8 3 7.5 3.4 3.0 2.1 10.8 4.9 4.3 14.6 6.6 5.8 4.0 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 - Stated X All Expected Counts are 1 or greater. √ No more than 20% of the Expected Counts are less than 5. X

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

Homework: Page 630: 17, 20, 21

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. 33.19 42.28 22.98 21.45 31.81 40.62 22.02 20.55 Definitely Would Mostly Likely Probably would Not Definitely Would Not Arkansas 35 45 28 12 Texas 30 38 17 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.

#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? Fish Consumption Total Subjects Prostate Cancer Never 124 14 Small part of diet 2621 201 Moderate part 2978 209 Large part 549 42 9.21 114.79 194.74 2426.26 221.26 2756.74 40.79 508.21 NO Prostate Cancer 14 110 201 2420 209 2769 42 507 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

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? Fish Consumption Total Subjects Prostate Cancer Never 124 14 Small part of diet 2621 201 Moderate part 2978 209 Large part 549 42 9.21 114.79 194.74 2426.26 221.26 2756.74 40.79 508.21 NO Prostate Cancer 14 110 201 2420 209 2769 42 507 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.

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. Political Affiliation Democrat Republican West 39 27 Northeast 35 15 Southeast 17 44 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 0 50 100 % of Dem. N NW SE Regional Location