Chapter 24 Chapter24 Presentation 1213 Comparing Means from Independent Samples Copyright 2009 Pearson Education Inc 1 Plot the Data A natural display for comparing two groups is boxplots of the data for the two groups placed sideby side For example Chapter24 Presentation 1213 Copyright 2009 Pearson Education Inc 2 Comparing Two Means Once we have examined the side by side boxplots of sample data we can turn to the comparison of the two population means The parameter of interest is the difference between the two population means 1 2 Chapter24 Presentation 1213 Copyright 2009 Pearson Education Inc 3 The Standard Error for Comparing Two Means For independent random quantities variances add standard deviations are not additive So the standard deviation of the difference between two sample means is SD y1 y2 12 n1 22 n2 We still don t know the true standard deviations of the two groups so we need to use the standard error s12 s22 SE y1 y2 n1 n2 Chapter24 Presentation 1213 Copyright 2009 Pearson Education Inc 4 Using t to Compare Two Means The sampling distribution of the difference in sample means of two independent groups is a Student s t The confidence interval we build is called a two sample t interval for the difference in means The corresponding hypothesis test is called a two sample t test Chapter24 Presentation 1213 Copyright 2009 Pearson Education Inc 5 Assumptions and Conditions Independence Assumption Each condition needs to be checked for both groups Randomization Condition Is the data from a random sample 10 Condition Were the samples for each group less than 10 of the respective populations Chapter24 Presentation 1213 Copyright 2009 Pearson Education Inc 6 Assumptions and Conditions cont Normal Population Assumption Nearly Normal Condition This must be checked for both groups A violation by either one violates the condition Independent Groups Assumption The two groups we are comparing must be independent of each other Chapter24 Presentation 1213 Copyright 2009 Pearson Education Inc 7 Two Sample t Interval When the conditions are met we are ready to find the confidence interval for the difference between means of two independent groups The confidence interval is y1 y2 t df SE y1 y2 where the standard error of the difference of the means is SE y1 y2 s12 s22 n1 n2 The critical value t depends on the particular confidence level C that you specify and on the number of degrees of freedom Chapter24 Presentation 1213 Copyright 2009 Pearson Education Inc 8 Degrees of Freedom The formula for the degrees of freedom for our t critical value is somewhat complex Because of this we will let technology calculate degrees of freedom for us See Degrees of Freedom Calculator for 2 Sample t Test from the Stat 201 help page or go to http web utk edu cwiek TwoSampleDoF Chapter24 Presentation 1213 Copyright 2009 Pearson Education Inc 9 Using JMP for Comparing Means Side by side box plots can be generated using the instructions in our JMP tutorials titled Sideby Side Box Plots A confidence interval and other output for the difference in two population averages can be generated using the instructions titled TwoSample t Procedure assuming unequal population variances Chapter24 Presentation 1213 Copyright 2009 Pearson Education Inc 10 Would You Pay More Buying a Nice Nearly New Camera from a Friend Example from p 625 X Seller Y Price Offered Interpret the interval 113 96 26 94 Chapter24 Presentation 1213 Copyright 2009 Pearson Education Inc 11 The Computational Details see output 211 42857 281 875 70 45 SE y1 y2 s12 s22 n1 n2 18 310321 2 46 432234 2 8 7 18 70566 see output Chapter24 Presentation 1213 Copyright 2009 Pearson Education Inc 12 The Computational Details cont df from the online calculator or JMP df 7 6229 t df The following online calculator will find t values for fractional degrees of freedom http www tutor homework com statistics tables statistics tables html See Misc Online Calculators from the Stat 201 Help Page So the t value for 95 confidence and 7 6229 df is 2 326 Chapter24 Presentation 1213 Copyright 2009 Pearson Education Inc 13 The Computational Details cont y1 y2 t df SE y1 y2 70 45 2 326 18 70566 70 45 43 509 113 96 1 2 26 94 see output Chapter24 Presentation 1213 Copyright 2009 Pearson Education Inc 14 The Computational Details cont Interpret each portion of the confidence interval calculation 70 45 2 326 18 70566 70 45 2 326 18 70566 Chapter24 Presentation 1213 Copyright 2009 Pearson Education Inc 15 In Class Example Facebook The file Ch24 Facebook contains a random sample of 14 freshmen out of 141 and 16 juniors out of 160 from survey data from Spring 09 Stat 201 classes Of those that use Facebook is there a difference between the average number of friends people have on Facebook between freshmen and juniors If you had to guess which group would you expect to have a higher average Chapter24 Presentation 1213 Copyright 2009 Pearson Education Inc 16 In Class Example Facebook cont Let s take a look at the side by side box plots of the results Does there look like a difference between the two groups What sort of differences do you see Chapter24 Presentation 1213 Copyright 2009 Pearson Education Inc 17 In Class Example Facebook Cont Check Condition 1 Randomization Condition Check Condition 2 10 Condition Check Condition 3 Nearly Normal Condition Check Condition 4 Independent Groups Chapter24 Presentation 1213 Copyright 2009 Pearson Education Inc 18 In Class Example Facebook Cont Results of checking the Nearly Normal Condition Freshman histogram Junior histogram Freshman goodness of fit test Junior goodness of fit test Chapter24 Presentation 1213 Copyright 2009 Pearson Education Inc 19 In Class Example Facebook Cont Results of the 2 sample t interval i e t Test in JMP What did JMP subtract from what Chapter24 Presentation 1213 Copyright 2009 Pearson Education Inc 20 In Class Example Facebook Cont Interpretation of the confidence interval and conclusion We can be confident that the difference between the number of friends people have on Facebook for juniors vs freshman in our database is between and Conclusion does the sample data provide evidence that in the population from which we sampled that junior freshman is different from zero Chapter24 Presentation 1213 Copyright 2009 Pearson Education Inc 21 In Class Example Facebook Cont In this case I know what the population means are calculated from the 141 freshman and 160 juniors in the original survey database junior is 560 75