1Chapter24 Presentation 1213Copyright © 2009 Pearson Education, Inc.Chapter 24 Comparing Means from Independent SamplesChapter24 Presentation 1213Copyright © 2009 Pearson Education, Inc.2Plot the Data A natural display for comparing two groups is boxplots of the data for the two groups, placed side-by-side. For example:Chapter24 Presentation 1213Copyright © 2009 Pearson Education, Inc.3Comparing 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 1213Copyright © 2009 Pearson Education, Inc.4The 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  We still don’t know the true standard deviations of the two groups, so we need to use the standard error:22121212SD y ynn 22121212ssSE y ynn Chapter24 Presentation 1213Copyright © 2009 Pearson Education, Inc.5Using t to Compare Two Means  The sampling distribution of the differencein 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 1213Copyright © 2009 Pearson Education, Inc.6Assumptions 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 1213Copyright © 2009 Pearson Education, Inc.7Assumptions 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 1213Copyright © 2009 Pearson Education, Inc.8Two-Sample t-IntervalWhen the conditions are met, we are ready to find the confidence interval for the difference between means of two independent groups. The confidence interval iswhere the standard error of the difference of the means isThe critical value t* depends on the particular confidence level, C, that you specify and on the number of degrees of freedom.22121212ssSE y ynn 12 12dfyytSEyy Chapter24 Presentation 1213Copyright © 2009 Pearson Education, Inc.9Degrees of Freedom The formula for the degrees of freedom for our tcritical 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/TwoSampleDoFChapter24 Presentation 1213Copyright © 2009 Pearson Education, Inc.10Using JMP for Comparing Means Side by side box plots can be generated using the instructions in our JMP tutorials titled “Side-by-Side Box Plots”. A confidence interval (and other output) for the difference in two population averages can be generated using the instructions titled “Two-Sample t Procedure (assuming unequal population variances)”.Chapter24 Presentation 1213Copyright © 2009 Pearson Education, Inc.11Would 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 1213Copyright © 2009 Pearson Education, Inc.12The Computational Details22121212ssSE y ynn =(46.432234)27(18.310321)28+= 211.42857 – 281.875 = -70.45(see output)= 18.70566(see output)Chapter24 Presentation 1213Copyright © 2009 Pearson Education, Inc.13The 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 So, the t* value for 95% confidence and 7.6229 dfis 2.326(See “Misc Online Calculators” from the Stat 201 Help Page)Chapter24 Presentation 1213Copyright © 2009 Pearson Education, Inc.14The Computational Details (cont.)12 12dfyytSEyy -70.45 ± 2.326 (18.70566)-70.45 ± 43.509(-113.96 < 1-2< -26.94)(see output)Chapter24 Presentation 1213Copyright © 2009 Pearson Education, Inc.15The Computational Details (cont.) Interpret each portion of the confidence interval calculation-$70.45 ± 2.326 ($18.70566) -$70.45 2.326 $18.70566Chapter24 Presentation 1213Copyright © 2009 Pearson Education, Inc.16In 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 1213Copyright © 2009 Pearson Education, Inc.17In 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 1213Copyright © 2009 Pearson Education, Inc.18In Class Example - Facebook (Cont.) Check Condition 1: Randomization Condition Check Condition 2: 10% Condition Check Condition 3: Nearly Normal Condition Check Condition 4: Independent GroupsChapter24 Presentation 1213Copyright © 2009 Pearson Education, Inc.19In Class Example - Facebook (Cont.) Results of checking the Nearly Normal Condition: Freshman histogram Junior histogram Freshman goodness of fit test Junior goodness of fit testChapter24 Presentation 1213Copyright © 2009 Pearson Education, Inc.20In 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 1213Copyright © 2009 Pearson Education, Inc.21In Class Example - Facebook (Cont.)