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# BASIC APPLIED STATISTICS Stat 0200

Pitt

GPA 3.52

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## Popular in Statistics

This 10 page Class Notes was uploaded by Josefa Cartwright Jr. on Monday October 26, 2015. The Class Notes belongs to Stat 0200 at University of Pittsburgh taught by Staff in Summer 2015. Since its upload, it has received 25 views. For similar materials see Basic Statistics in Statistics at University of Pittsburgh.

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Date Created: 10/26/15

C 2007 Nancy Pfenning 7 E l l l Looking Back Review Lecture 23 El 4 Stages of Statistics I Data Production discussed in Lectures 14 Inference for Categorical Variable I Displaying and Summarizing Lectures 542 More About Hypothesis Tests I Probability discussed in Lectures 1320 I Statistical Inference Examples of Tests With 3 Forms of Alternative El 1 categorical con dence intervals hypothesis tests EIHOW Form of Alternative Affects Test 1 quantitative ElWhen P Value is Small Statistical Significance ElHypothesis Tests in LongRun ElReIating Test Results to Confidence Interval categorical and quantitative 2 categorical 2 quantitative e 2mm Nancy Pfennan Elementary etatlstles Leeklng attne ale F39leture e 2mm Nancy Pfennan Elementary etatlstles Leeklng attne ale F39leture L23 2 l l 39 7 Three Types of Inference Problem Review Hypothesis Test About p Review In a sample of 446 students 055 ate breakfast State null and alternative hypotheses H o and H a 1 What is our best guess for the proportion of all Null is status quo alternative rocks the boat students who eat breakfast p gt 100 Hoppo VS Ha pltpo P01ntEst1mate p 72 390 2 What interval should contain the proportion of 1 l Consider sampling and study designl all Students Who eat breakfaSt 2 Summarize with standardize to Z assuming Confidence Interval that H0 p 2 390 is true consider if Z is large 3 Do more than half 50 of all students eat 3 Find Pvalueprobof Z this far abovebelowaway breakfast from 0 consider if it is small Hypothesis Test 4 Based on size of Pvalue choose H 0 or H a e 2mm Nancy Pfennan Elementary etatlstles Leeklng attne ale F39leture L23 3 e 2mm Nancy Pfennan Elementary etatlstles Leeklng attne ale F39leture L23 4 Elementary Statistics Looking at the Big Picture 1 C 2007 Nancy Pfenning Checking Sample Size Cl vs Test Example Checking Sample Size in Test ll Confidence Interval Require observed counts El BaCkgl Ollndi 304000075 Students PiCked 7 at in and out of category of interest to be at least random from 1 to 20 Want to test H 0 3172005 VS 10 13 X 2 10 Haiflgt005i 1 h f d nu 13 n X 2 10 El Ques 10n sn arge enoug tOJHStlfy in mg PValue based on normal probabilities II Hypothes1s Test Require expected counts in and out of category of interest to be at least 10 assume p p0 TWO 2 10 n1 190 Z 10 e 2mm Nancy F39fErlrllrlg Elementary Statistles Luuklng attne Big F39lcture L23 5 e 2mm Nancy F39fErlrllrlg Elementary Statistles Luuklng attne Big F39lcture L23 6 i l 39 a Example Checking Sample Size in Test Example T est with gt Alternative Review I Background 304000075 students picked 7 at El Note Step 1 requires 3 checks random from 1 to 20 Want to testHO 72005 vs I Is sample unbiased Sample proportion has mean 005 Ha pgt005 I Is population 210n Formula for sd correct R o I Are npo and nlpo both at least 10 Find or estimate D esponse39 PValue based on normal probabilities quotp02 1 Students are typical h 4 39 u i issue at hand FIGPO 2 pr005 sd of 13 is 005l 005nd o 75 quot Looking Back For con dence interval checked 2 39 30 and 370 both at least 10 W 3 PValue PZ Z 229 is smalljust over 001 4 RejectHO conclude Ha picks were biased for 7 e 2mm Nancy F39fErlrllrlg Elementary Statistles Luuklng attne Big F39lcture L23 8 e 2mm Nancy F39fErlrllrlg Elementary Statistles Luuklng attne Big F39lcture L23 3 Elementary Statistics Looking at the Big Picture 2 C 2007 Nancy Pfenning Example Test with Less Than Alternative Example Test with Less Than Alternative I Background 111230 of surveyed commuters at a El Background 111230 of surveyed commuters at a university walked to school uanerSlty walked to SCh001 II Question Do fewer than half of the university s D Response FlrSt Wnte H03 VS Ha commuters walk to school 1 Students need to be representative in terms of year 2 Output9 33 2 Test and CI for One Proportion Test and CI for One Proportion Test ofp05 vsplt05 Test ofpO5 vsplt05 Sample X N Sample p 9507 Upper Bound ZValue P Value Sample X N Sample p 9507 Upper Bound ZValue PValue 1 111 230 0482609 0536805 053 0299 1 111 230 0482609 0536805 053 0299 3 Pvalue 4 RejectHo c 2mm Nancy Pfenning Eiementary Statistics Luuking atthe Big Picture L23 in c 2mm Nancy Pfenning Eiementary Statistics Luuking atthe Big Picture L23 i3 Example Test with Less Than Alternative Example Test with Not Equal Alternative CI Note Pvalue is a lefttailed probability because I Background 43 of Florida s community college alternative was less than students are disadvantaged II Question Is disadvantaged at Florida Keys Community College 169356475 unusual Test and CI for One Proportion Test of p 043 vs p not 043 Sample X N Sample p 950 CI Z Value P Value 1 169 356 0474719 0422847 0526592 170 0088 o 2mm Nancy Pfenning Eiementary Statistics Luuking atthe Big Picture L23 M o 2mm Nancy Pfenning Eiementary Statistics Luuking atthe Big Picture L23 i5 Elementary Statistics Looking at the Big Picture 3 C 2007 Nancy Pfenning Example Test with Not Equal Alternative Example Test with Not Equal Alternative 393 BaCkgmundi 43 0f Florida s community COllege CI Note Pvalue is a twotailed probability because students are disadvantaged alternative was not equal El Response First write H 0 vs H a 1 356043 3561043 both210 pop210356 2 p z Test and CI for Cine Proportion Test of p 043 vs p not 043 Sample X N Sample p 9507 CI Z Value P Value 1 169 356 0474719 0422847 0526592 170 0088 3 Pvalue 4 Reject H 0 c 2007 Nancy Prennirig Eiernentaiy Statistics Luuking atthe Big Picture L23 i7 c 2007 Nancy Prennirig Eiernentaiy Statistics Luuking atthe Big Picture L23 i9 90959899 Rule Outside Probabilities Onesided or Twosided Alternative I Form of alternative hypothesis impacts 05 Pvalue I Pvalue is the deciding factor in test Pval e just unde area025 1 area0 l area 005 area025 I Alternative should be based on what researchers hopefear suspect is true 3005 before snooping at the data area01 i i i 4545 I 70 just gm 1 6459 L1 645 I Z I If lt or gt is not obv1ous use twoSided 4960 l I 2325 1 329326 alternative more conservatlve 72576 2576 e mi Nancy Prenning Eiementaiy Statistics Luuking atthe Big Picture L23 In e mi Nancy Ptenning Eiementaiy Statistics Luuking atthe Big Picture L23 2i Elementary Statistics Looking at the Big Picture 4 C 2007 Nancy Pfenning E Example How F arm of Alternative A ects Test Example How F arm of Alternative A ects Test I Background 43 of Florida s community college El Background 43 of Florida s community college students are disadvantaged students are disadvantaged II Question Is disadvantaged at Florida Keys 539 Response NOW Wme H03 VS H03 169356475 unusually high 1 Same checks of data production as before 2 Same 0475z170 Test of p 043 vs p gt 048 Sample X N Sample p 9501 Lower Bound Z Value PValue 1 169 356 0474719 0431186 170 0044 3 Now Pvalue 4 Reject H 0 e 2mm Nancy Ptenning Eiernentary Statistics Luuking attne Big Picture L23 23 e 2mm Nancy Ptenning Eiernentary Statistics Looking attne Big Picture L23 25 i i 39 e Pvalue for One or TwoSided Alternative Thinking About Data I Pvalue for onesided alternative is half Before getting caught up in details of test Pvalue for twosided alternative consider evidence at hand I Pvalue for twosided alternative is twice Pvalue for onesided alternative For this reason twosided alternative is more conservative larger Pvalue harder to reject Ho e 2mm Nancy Ptenning Eiernentary Statistics Luuking attne Big Picture L23 2B e 2mm Nancy Ptenning Eiernentary Statistics Luuking attne Big Picture L23 27 Elementary Statistics Looking at the Big Picture 5 C 2007 Nancy Pfenning Example Thinking A bout Data at Hand El Background 43 of Florida s community college students are disadvantaged At Florida Keys the rate is 475 Question Is the rate at Florida Keys signi cantly lower El 2mm mnwmm Eiemenuwsuusucs mm tithe swim 1232s Example Thinking About Data at Hand El Background 43 of Florida s community college students are disadvantaged At Florida Keys the rate is 475 El Response 2mm mm mm ammw Stalslics mm um aw mm m cm De nition alpha 01 cutoff level which signi es a Pvalue is small enough to reject H 0 Eiemenuwsuusucs mm tithe swim 12331 How Small is a Small PValue Elementary Statistics Looking at the Big Picture I Avoid blind adherence to cutoff 05005 I Take into account 1 Past considerations is 10 Written in stone or easily subject to debate Future considerations What would be the consequences of either type of error I Rejecting H0 even though it s true I Failing to reject He even though it s false I Consider decisions encountered so far U 2mm mm mm ammw Stalslics mm um aw mm m 32 C 2007 Nancy Pfenning Example Reviewing P values and Conclusions El Background Consider our prototypical examples I Are random number selections biased PvaluF001 l I Do fewer than half of commuters walk Pvalue4299 I Is disadvantaged signi cantly different PvaluF0088 I Is disadvantaged signi cantly higher Pvalue0044 El Question What conclusions did we draw based on those Pvalues 2mm mnwmm amnuwsmsm mm mm aiwmme 1233 Example Reviewing P values and Conclusions El Background Consider our prototypical examples I Are random number selections biased Pvalue4011 I Do fewer than half of commuters walk Pvalue0299 I Is disadvantaged signi cantly different Pvalue4088 I Is disadvantaged signi cantly higher Pvalue0044 El Response Consistent with 005 as cutoff Oi I P value001 l 9Rej ect 7 I P value02999 Reject 7 I P value0088 9Rej ect 7 I P value0044 9Rej ect 7 2mm mm mm gummy Statstics mm um aw mm m 35 Example CutO s for Small quotP Value El Background Bookstore chain will open new store in a city if there s evidence that its proportion of college grads is higher than 026 the national rate El Question Choose cutoff 010 005 001 I if no other info is provided I if chain is enjoying considerable pro ts owners are eager to pursue new ventures if chain is in financial difficulties can t afford losses if unsuccessful due to too few grads 2mm mnwmm amnuwsmsm mm mm aiwmme mas Example CutO s for Small quotP Value Elementary Statistics Looking at the Big Picture El Response Choose cut0ff010 005 001 if no other info is provided El use77 I if chain is enjoying considerable pro ts owners are eager to pursue new ventures El use 7 I if chain is in financial difficulties can t afford loss if unsuccessful due to too few grads El use 2mm mm mm gummy Statstics mm um aw mm m as C 2007 Nancy Pfenning De nition Role of Sample Size n Statistically significant data produce Pvalue small enough to rejectHo Z plays a role l Large 11 may reject H 0 even though observed proportion isn t very far frompo Z 13 290 2 from a practical standpoint iPo1Po p01P0 TL Reject Ho ifPvalue small if Z large if I Sample proportion 13 far from p0 Very small Pvalue strong evidence against Ho but p not necessarily very far from po l Small 11 may fail to reject H 0 even though I Sample size n large it is false I Standard deviation small if pois close to 0 or 1 Failing to reject false H0 is 2 type of error e 2mm Nancy F39fErlrllrlg Elementary Statistles Leeklng attne Big F39lcture L23 3a e 2mm Nancy F39fErlrllrlg Elementary Statistles Leeklng attne Big F39lcture L23 4n i l De nition Hypothesis Test and LongRun Behavior I Type I Error reject null hypothesis even Repeatedly carry out hypothesis tests of p05 though it is true false positive based on 20 coinflips using cutoff 5 i Probability is cutoff Ct In the long run 5 of the tests will reject I Type 11 Error fail to reject null HO p05 even though it s true hypothesis even though it s false false negative e 2mm Nancy F39fErlrllrlg Elementary Statistles Leeklng attne Big F39lcture L23 M e 2mm Nancy F39fErlrllrlg Elementary Statistles Leeklng attne Big F39lcture L23 42 Elementary Statistics Looking at the Big Picture 8 C 2007 Nancy Pfenning HypotheSIS Test and LongRun Behav10r Confidence Interval and Hypothesis Test Results 20 mi lps test H0 pgggy sb gggggg equal 50 l Con dence Interval range of plausible values TlTITHTH39lTHHT 7 HH 39 Wopmmfheadsg z45pvaiiesss I Hypothesis Test decides if a value is plausible HTI HHTHH ITI HTHTFTHHT Proportion of head 40 2389 p39Vame39371 4 IIIfOImally Z iagvaalueaaw 4 El If 170 is in confidence interval don t re ect Ho 7170 39 El pr0 is out51de confidence interval reject Ho 7170 THHHl tTHHHTHT HHH Z 2 24pvaiue 025 Relationship between 95 confidence interval pr p lhead 3975 l and twosided test with 05 as cutoff for pvalue 0 llips oi 20 o 95 chests do not reject Ho If 0 IS here I39e eCt HO 15 39 5 oi tests reject Ho i i l Cl 95 confidence interval i for population proportion V V Tl39H HTTTHTTHHTHHH proportion of heads 8204O 2289 prvalue37t A do not reiect Ho 0 2mm Nancy Pfenning Elementary Statistics Looking atthe Big Picture L23 43 0 2mm Nancy Pfenning I If p0 is here do not relem Ho pjpo I L23 44 Example Test Results Based on C Example Test Results Based on C I Background A 95 confidence interval for I Background A 95 confidence interval for proportion of all students choosing 7 at proportion of all students choosing 7 at random from numbers 1 to 20 is random from numbers 1 to 20 is 0055 0095 0055 0095 I Question Would we expect a hypothesis test I Response to reject the claim p005 in favor of the claim pgt005 Elementary Statistics Looking at the Big Picture 9 Example CI Results Based on Test El Background A hypothesis test did not reject HO p0 5 in favor of the alternative H 11 plt05 El Question Do we expect 05 to be contained in a con dence interval for p 2mm mnwmm amnuwsmgm mm tithe swim mm C 2007 Nancy Pfenning Example CI Results Based on Test El Background A hypothesis test did not reject HO p05 in favor of the alternative Ha plt05 El Response 2mm mm mm ammw Statstics mm um aw mm m 5 Lecture Summary iiI ore Hypothesis Tests for Proportions El Examples with 3 forms of alternative hypothesis El Form of alternative hypothesis I Effect on test results I When data render formal test unnecessary I Pvalue for lsided vs 2sided alternative Cutoff for small Pvalue Statistical signi cance role of n Type I or H Error Hypothesis tests in longrun EIEIEIEI Relating tests and confidence intervals 2mm mnwmm amnuwsmgm mm tithe swim Liam Elementary Statistics Looking at the Big Picture

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