Probability and Statistics AMP Institutes & Workshops Saturday, April 4 th, 2015 Trey Cox. Ph. D. Mathematics Faculty Chandler-Gilbert Community College James Spiker A.P. Statistics Basha High School, Chandler, AZ This work was supported in part by MSP grant # through the National Science Foundation. Opinions expressed are those of the authors and not necessarily those of the NSF.

© 2014 Relay Graduate School of Education and Teach For America. All rights reserved. 2 On April 15, 1912, the Titanic struck an iceberg and rapidly sank with only 710 of her 2,204 passengers and crew surviving. I wonder…were the rich people more likely to survive? Was chivalry alive and well on the Titanic? Was it “every man for himself”? Bivariate Data Analysis – Qualitative/Categorical

© 2014 Relay Graduate School of Education and Teach For America. All rights reserved. 3 SurvivedDid not survive 1 st class passengers nd class passengers rd class passengers Bivariate Data Analysis – Qualitative/Categorical

© 2014 Relay Graduate School of Education and Teach For America. All rights reserved. 4 SurvivedDid not survive 1 st class passengers nd class passengers rd class passengers Bivariate Data: Passenger and Survival Explanatory variable? ______________ Responsory variable? _______________ Bivariate Data Analysis – Qualitative/Categorical

© 2014 Relay Graduate School of Education and Teach For America. All rights reserved. 5 Bivariate Data Analysis – Qualitative/Categorical CCSS.MATH.CONTENT.8.SP.A.4 Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables.

© 2014 Relay Graduate School of Education and Teach For America. All rights reserved. 6 Bivariate Data Analysis – Qualitative/Categorical CCSS.MATH.CONTENT.HSS.ID.B.5 Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data. CCSS.MATH.CONTENT.HSS.CP.A.4 Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities.

© 2014 Relay Graduate School of Education and Teach For America. All rights reserved. 7 SurvivedDid not survive 1 st class passengers nd class passengers rd class passengers Bivariate Data Analysis – Qualitative/Categorical SurvivedDid not surviveTOTAL 1 st class passengers nd class passengers rd class passengers TOTAL Two-way table marginal totals Is there an association between class and survival?

© 2014 Relay Graduate School of Education and Teach For America. All rights reserved. 8 Bivariate Data Analysis – Qualitative/Categorical SurvivedDid not survive TOTAL 1 st class passengers nd class passengers rd class passengers TOTAL So, what do you think?: Is there an association between class and survival?

© 2014 Relay Graduate School of Education and Teach For America. All rights reserved. 9 Bivariate Data Analysis – Qualitative/Categorical SurvivedDid not surviveTOTAL 1 st class passengers nd class passengers rd class passengers Which table is easier to use to come to a conclusion? Why? What is the difference between the two tables? How is the second table generated from the first table? SurvivedDid not surviveTOTAL 1 st class passengers nd class passengers rd class passengers Relative frequency table

© 2014 Relay Graduate School of Education and Teach For America. All rights reserved. 10 SurvivedDid not survive 1 st class passengers nd class passengers rd class passengers3665 TOTAL100 How was this second table generated? Bivariate Data Analysis – Qualitative/Categorical SurvivedDid not surviveTOTAL 1 st class passengers nd class passengers rd class passengers What question do the two tables help you answer?

© 2014 Relay Graduate School of Education and Teach For America. All rights reserved. 11 Bivariate Data Analysis – Qualitative/Categorical SurvivedDid not survive Children in 1 st class41 Women in 1 st class1394 Men in 1 st class58118 Children in 2 nd class220 Women in 2 nd class8312 Men in 2 nd class13154 Children in 3 rd class3050 Women in 3 rd class9188 Men in 3 rd class60390 Your Turn! Can you make any substantiated claims from this data?

© 2014 Relay Graduate School of Education and Teach For America. All rights reserved. 12 Q: Is there a more formal way to quantitatively measure if there is a significant difference between the different classes or genders in terms of who was saved and who perished? A: The chi-square test provides a method for testing the association between the row and column variables in a two-way table. Where does all of this go?

© 2014 Relay Graduate School of Education and Teach For America. All rights reserved. 13 The expected value for each cell in a two-way table is equal to: where n is the total number of observations included in the table. Q: Why does this formula make sense for calculating the expected value (i.e. what we would expect the table values to be if there were no association)? In other words, why would we multiply the row total and column total and divide by n? SurvivedDid not surviveTOTAL 1 st class passengers nd class passengers rd class passengers TOTAL Where does all of this go?

© 2014 Relay Graduate School of Education and Teach For America. All rights reserved. 14 Expected Values? SurvivedDid not surviveTOTAL 1 st class passengers nd class passengers rd class passengers TOTAL SurvivedDid not survive 1 st class passengers 2 nd class passengers 3 rd class passengers Where does all of this go? SurvivedDid not survive 1 st class passengers nd class passengers rd class passengers

© 2014 Relay Graduate School of Education and Teach For America. All rights reserved Explain why the calculation of the chi-square statistic makes sense as a way to quantify if there is a difference between the variables in a table. 1.Do you think a large or small chi-square value would indicate an association between the two categorical variables? Explain. Where does all of this go?

© 2014 Relay Graduate School of Education and Teach For America. All rights reserved. 16 CCSS.MATH.CONTENT.HSS.CP.A.3 Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B. CCSS.MATH.CONTENT.HSS.CP.B.6 Find the conditional probability of A given B as the fraction of B's outcomes that also belong to A, and interpret the answer in terms of the model. CCSS.MATH.CONTENT.HSS.CP.B.7 Apply the Addition Rule, P(A or B) = P(A) + P(B) - P(A and B), and interpret the answer in terms of the model. CCSS.MATH.CONTENT.HSS.CP.B.8 Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(B|A) = P(B)P(A|B), and interpret the answer in terms of the model. Where does all of this go?

© 2014 Relay Graduate School of Education and Teach For America. All rights reserved. 17 Conditional Probability – The Power of Two-way Tables 1.If one of the passengers is randomly selected, what is the probability that this passenger was in first class? third class? 1.If one of the passengers is randomly selected, what is the probability that this passenger was in the first class and survived? 1.If one of the passengers who survived is randomly selected, what is the probability that this passenger was in third class? Where does all of this go?

© 2014 Relay Graduate School of Education and Teach For America. All rights reserved. 18 Have You Seen a Probability Problem like this one?... The probability that a person has a certain virus is A test used to detect the virus in a person is positive 80% of the time if the person has the virus and 5% of the time if the person does not have the virus. Let A be the event that “the person is infected” and B be the event “the person tests positive”. If a person tests positive, what is the probability that the person has the virus?

© 2014 Relay Graduate School of Education and Teach For America. All rights reserved and solved this way?

© 2014 Relay Graduate School of Education and Teach For America. All rights reserved. 20 Why not solve it like this?... Use a contrived frequency table Test +Test -TOTAL Has virus415 Does not have virus TOTAL The probability that a person has a certain virus is A test used to detect the virus in a person is positive 80% of the time if the person has the virus and 5% of the time if the person does not have the virus. Let A be the event that “the person is infected” and B be the event “the person tests positive”. If a person tests positive, what is the probability that the person has the virus?