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PSYS 04, Statistics for Psychological Science: 2/18/16

by: Delaney Row

PSYS 04, Statistics for Psychological Science: 2/18/16 PSYS 054

Marketplace > University of Vermont > Psychlogy > PSYS 054 > PSYS 04 Statistics for Psychological Science 2 18 16
Delaney Row
GPA 3.65

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About this Document

These notes cover the week of 2/15/16, with topics on z-test and t-test
Statistics for Psychological Science
Keith Burt
Class Notes
PSYS 054, Psych stats, Statistics for Psychological Science
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This 10 page Class Notes was uploaded by Delaney Row on Saturday February 20, 2016. The Class Notes belongs to PSYS 054 at University of Vermont taught by Keith Burt in Fall 2016. Since its upload, it has received 15 views. For similar materials see Statistics for Psychological Science in Psychlogy at University of Vermont.


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Date Created: 02/20/16
PSYS  054  –  Statistics  For  Psychological  Science   Notes  for  the  week  of  2/15/16  –  2/19/16       2/16/16     Directional  vs.  Non-­‐directional:  Decision  Tree     • With  directional,  you  have  to  predict  which  direction  it  will  be   o Positive  –  target  sample  mean  would  be  higher   o Negative  –  target  sample  mean  would  be  lower   • Remember  this  is  all  based  on  the  researchers  hypothesis,  not  on  the   data!     Quantifying  the  Probability  of  Mistakes:  Alpha  and  Beta   • NOTE:  we  never  know  the  true  state  of  nature   • Alpha  is  under  type  I  error,  beta  is  under  type  II  error     Two-­‐Tailed  Test  Setup     • If  our  null  hypothesis  is  true,  there  is  a  low  probability  of  getting   sample  means  in  the  two  shaded  gray  regions  (rejection  regions),  and   a  high  probability  of  getting  sample  means  that  are  in  the  middle   white  region   • By  setting  this  cutoff,  we  are  saying  that  if  we  get  sample  means  that   fall  in  the  gray  regions,  then  they  must  be  the  result  of  an  error     Setting  Rejection  Regions   • The  area  under  the  curve  of  rejection  region  is  called  “alpha  level”  (α),  and  it   is  expressed  as  a  probability     o Two  tailed  distribution  =  the  sum  of  the  area  under  both  extremes   o One  tailed  distribution  =  the  area  under  one  extreme   o The  score  that  falls  at  the  alpha  level  is  called  the  critical  value     • We  will  set  the  cutoff  at  5%  (this  means  there  is  a  5%  chance  of  making  a   type  I  error)   o Two  tailed  =  the  cutoff  will  be  at  the  most  extreme  low  2.5%  and  high   2.5%   o One  tailed  =  the  cutoff    will  be  the  most  extreme  5%  (either  on  the  left   or  right)   • A  one  tailed  test  gives  you  more  power  because  the  5%  is  all  located  on  one   side     One  Tailed  Test  Setup  (positive  direction):  Alpha  of  0.05     • This  is  still  a  distribution  of  sample  means     • All  the  5%  (rejection  region)  will  be  on  the  right  hand  side  because  this  is  a   one  tailed  test  in  the  positive  direction     • We  would  reject  our  null  hypothesis  if  our  target  sample  mean  is  under  the   rejection  area     Directional  vs.  Non-­‐directional  Hypothesis   • Let  μ  be  the  mean  of  whatever  population  our  sample  comes  from   • SAT  prep  example  (500  was  non-­‐special  mean):   o Our  expectation:  Kaplan  classes  improves  scores  =  directional  and   positive…   § H 0  μ  =  500   § H 1  μ  >  500   • Non-­‐directional  example:   o GPAs  are  significantly  different     § Shows  it  will  be  a  two  tailed  test  (either  higher  OR  lower)   § H 0  μ  =  2.97   § H 1  μ  ≠  2.97  (this  “does  not  equal”  sign  is  characteristic  of  a  two   tailed  test)     The  Story  So  Far…   1. Determine  if  it  will  be  a  directional  or  non-­‐directional  test   2. Set  alpha  level  (for  our  purpose  it  is  .05)   3. Create  sampling  distribution   4. Find  critical  values  to  draw  rejection  region  (depends  on  one  tail  or  two  tail)   5. Reject  or  retain  the  null  hypothesis  based  on  whether  the  target  sample  is  in   rejection  region     Central  Limit  Theorem   • The  central  limit  theorem  tells  you  what  the  sampling  distribution  looks  like   (includes  the  mean  and  SD  of  sampling  distribution)  with  just  knowing  mean,   SD  of  population,  and  N   • It  creates  sampling  distribution  of  the  mean  without  actually  taking  10,000+   random  samples   2 • Given  a  population  with  mean  μ  and  variance  σ ,  the  sampling  distribution  of   the  mean  will:   o Have  a  mean  equal  to  μ   o Have  a  variance  equal  to  σ  /  N    (and  SD  equal  to  σ  /  √N)   o Approach  a  normal  distribution  as  N  increases   • The  central  limit  theorem  also  states  that  the  shape  is  essentially  normal  as   long  as  N>30  (it  will  get  more  normal  the  bigger  the  sample  is)   • This  theorem  underlies  formulas  on  most  statistical  tests  –  very  important!     Central  Limit  Theorem:  Notation   • μ  sub  xbar  =  the  mean  of  the  sampling  distribution  of  the  mean   • σ  sub  xbar  =  the  standard  deviation  of  the  sampling  distribution  of  the  mean   (also  called  the  ‘standard  error’)     Z-­‐test   • The  z-­‐test  makes  use  of  everything  we  have  used  so  far   • NOTE:  the  z-­‐test  is  different  than  the  z-­‐score   • Z-­‐test  is  useful  when:   o You  want  to  compare  a  target  sample  mean  to  a  population  mean   o You  know  the  population  mean   o You  know  the  population  SD   • Basic  idea:   o  Convert  observed  sample  mean  to  a  Z-­‐score  (=location  on  the   sampling  distribution)  using  the  Central  Limit  Theorem   o Find  probability  of  getting  that  Z-­‐score  using  area  under  the  curve   (Appendix  E.10  in  the  textbook)     Z-­‐test  Formula     • Sample  mean  minus  the  population  mean,  all  divided  by  the  standard  error   o Standard  error  =  populations  SD  divided  by  the  square  root  of  N     Z-­‐test  Example:  SAT  Data   • N  =  30  sample  of  high  school  students  taking  an  SAT  prep  class;  sample  mean   of  540   • What  we  know:  population  of  students  NOT  taking  a  prep  class  has  μ  =  500,  σ   =  100   • We  can  plug  these  4  numbers  into  the  z  statistic  formula   • This  is  a  one  tailed  test  (hypothesizing  increase  in  scores)   o Research  question:  does  taking  a  prep  class  increase  SAT  scores?     Z-­‐test:  Basic  Steps   • We  have  to  figure  out  where  rejection  regions  will  be:   1. Establish  alpha  level,  rejection  region,  and  critical  values  (here,  critical  value   is  a  Z-­‐score)   2. Find  the  standard  error  of  the  population   3. Convert  observed  sample  mean  to  Z-­‐score   4. Compare  your  calculated  Z-­‐score  to  the  critical  value   5. Reject  or  retain  the  null  hypothesis     Critical  Values  of  z   • Appendix  E.10  in  the  textbook   • In  a  one-­‐tailed  test,  we  need  to  find  the  z-­‐score  that  “cuts  off”  the  upper  5%  of   the  normal  distribution   • We  have  to  find  a  z  score  that  has  exactly  5%  of  the  area  above  it   o Use  positive  z  score   o There  is  no  z  score  exactly  at  .05  so  we  take  the  two  surrounding  it…z   score  of  1.65   o Anything  above  1.65  =  in  rejection  region   • If  our  test  was  a  one  tailed  negative,  we  would  use  z  score  of  -­‐1.65  and  reject   anything  that  fell  below  -­‐1.65       Critical  Values  of  z   • In  a  two-­‐tailed  test,  we  need  to  find  the  z-­‐score  that  “cuts  off”  the  upper  (and   lower)  2.5%  on  each  side  of  the  normal  distribution   • We  do  the  same  thing  we  did  for  the  one  tailed  test  with  the  Appendix  E.10   and  we  get  a  z-­‐score  of  +/-­‐  1.96       Critical  Values  of  z   • These  critical  values  never  change  for  a  z-­‐test  no  matter  what  –  memorize   them!   o +/-­‐  1.96  for  two  tailed  tests   o 1.65  (negative  or  positive,  but  never  both)  for  one  tailed  tests     Z-­‐test  of  SAT  Study:  Step-­‐by-­‐Step   • In  a  directional  test,  alpha  =  .05,  therefore  critical  value  is  z-­‐score  that  cuts  off   upper  5%  of  distribution   o From  previous  slides:  this  is  +1.65   o If  we  observe  any  z-­‐score  greater  than  1.65,  reject  null  hypothesis…   otherwise  we  retain  it   • Population  standard  error  =  (σ/√N)  =     o 100  /  √30      =  100  /  5.48      =  18.25   • Using  formula:    Z  =    (540  -­‐  500)  /  18.25    =   o 40  /  18.25  =  2.19   • Comparing  observed  z-­‐score  and  critical  value:   o 2.19    >  1.65,  therefore….   • Reject  the  null  hypothesis!     • NOTE:  for  our  purposes,  if  you  get  a  z-­‐score  that  is  exactly  at  the  critical   value…reject  it     One-­‐Sample  t-­‐test   • The  z-­‐test  depends  on  a  lot  of  information  from  the  population,  which  isn’t   always  realistic   • The  t-­‐test  is  for  situations  in  which  the  population  variance  is  unknown   • T-­‐test  is  one  of  the  most  common  statistical  tests  used  today   • There  are  different  versions  of  this  test  now   2/18/16     One-­‐Sample  t-­‐test   • T-­‐test  is  used  for  circumstances  in  which  the  population  SD  is  unknown   o Most  of  the  time  we  do  not  know  the  population  SD  anyway   • In  the  one-­‐sample  t-­‐test,  we  are  comparing  a  sample  mean  to  a  hypothesized   population  value     T-­‐test  Information   • What  you  need  for  a  one-­‐sample  t-­‐test:   o Target  sample  mean   o Target  sample  SD   o Population  mean  assuming  the  null  hypothesis  H   0 o Target  sample  size   • We  are  computing  a  t-­‐statistic  in  a  similar  way  as  the  Z-­‐statistic   • BUT  we  are  replacing  the  sample  SD  with  what  was  the  population  SD  (we   don’t  have  to  know  the  population  SD)     T-­‐test  formula       Z-­‐test  vs.  t-­‐test   •  NOTE:  the  critical  values  1.65  and  +/-­‐  1.96  work  ONLY  for  z-­‐test  (when  you   know  the  population  SD)   Z-­‐test   t-­‐test         Known  population  M,  SD   Hypothesized  population  M,     unknown  SD       Denominator  is  population  SD  /  √N   Denominator  is  sample  SD  /  √N       Can  compare  calculated  z-­‐value  with   CAN’T  use  Appendix  E.10  to   critical  value  using  normal   compare  t-­‐value  with  critical  value       distribution  table  (Appendix  E.10)         More  Z-­‐test  and  t-­‐test   • Z-­‐test  requires  known  μ  and  σ  (SD)   • One-­‐sample  t-­‐test  allows  unknown  σ   o Hypothesized  μ  (under  H )   0   o σ  is  estimated  using  s,  the  sample  SD   • Because  of  estimation,  cannot  use  Z-­‐distribution   •  “Student’s”  solution:  created  a  t-­‐distribution   o Actually  many  distributions,  depending  on  sample  size   • T  test  causes  for  some  error  (because  you  are  estimating)   • T  test  makes  it  so  the  sampling  distribution  is  no  longer  normally   distributed…it  takes  on  a  new  shape   o This  new  distribution  we  call  a  t  distribution…it  still  looks  fairly  like  a   normal  distribution   § As  the  sample  size  gets  larger  and  larger,  the  t  distribution   looks  more  and  more  like  a  normal  distribution  (because  its   getting  closer  to  a  population)   § As  sample  size  gets  small,  t  distribution  starts  getting   platykurtic  (flat,  pushed  down)     Test  Assumptions   • The  Z-­‐test  assumes:   o The  data  are  independently  sampled  from  a  normally  distributed   population   o Mean  (μ)  and  standard  deviation  (σ)  are  known   • T-­‐test  assumes   o The  data  are  independently  sampled  from  a  normally  distributed   population     Degrees  of  Freedom   • With  the  t-­‐test,  we  are  estimating  a  population  mean  based  on  the  sample   mean   o We  have  to  account  for  error!   • Degrees  of  freedom  are  a  quality  of  any  statistical  test   o It  applies  to  every  test  we  do  except  for  the  z-­‐test   o It  is  a  correction  factor   • We  know  deviations  from  sample  mean  sum  to  zero   • So,  given  a  particular  sample  mean,  only  N-­‐1  observations  are  really  “free”  to   vary   • We  give  our  statistical  test  N-­‐1  degrees  of  freedom     T-­‐test:  Steps   1. Establish  what  information  you  have   2. Decide  if  the  t-­‐test  is  appropriate…if  it  is,     3. Calculate  your  t-­‐statistic     4. Calculate  your  degrees  of  freedom  (using  N-­‐1)   5. Use  Appendix  E.6  in  the  textbook  to  find  critical  values   6. Compare  your  t-­‐statistic  to  critical  value   a. If  your  t  statistic  is  more  extreme  than  the  critical  value,  then  reject   the  null  hypothesis  (if  not,  retain  the  null  hypothesis)     Using  Appendix  E.6  in  The  Textbook     • The  column  you  pick  depends  on:   o Your  alpha  level  (level  of  significance)   o Whether  your  test  is  one  tailed  or  two  tailed   • In  this  class  we  will  almost  always  be  using  the  alpha  level  0.05   • When  picking  a  row,  this  depends  on  your  degrees  of  freedom  (DF)   o For  a  one  sample  t  test,  it  is  N-­‐1   • NOTE:  one  issue  is  you  might  have  to  add  a  negative  sign  or  a  plus  or  minus   sign  to  the  critical  value  of  t,  based  on  your  distribution,  it  if  it  negative  or   two  tailed  respectively       T-­‐test  Example:  Developmental  Milestones   • Doctors  use  standard  charts  to  measure  the  progress  of  children’s   development  during  routine  check-­‐ups   o Children  average  15-­‐word  vocabularies  by  age  18  months   o Target  sample  (N=24):  walked  early  and  ate  solid  foods  early   o Prediction  to  test:  they  are  talking  early  as  well   § We  have  attained  15-­‐word  vocabulary  at  M  =  age  16.5  months,   SD  =  1  month   • We  want  to  know  if  this  talking  early  sample  of  children  sample  mean  is   statistically  significant     T-­‐test:  Set  Hypotheses   • This  example  is  DIRECTIONAL   • Setting  up  the  null  (0 )  and  alternative  (H 1  hypotheses   o H  0  sample  drawn  from  population  with  15-­‐word  vocabulary  at  mean   of  18  months   § H :0  μ  =  18   o H  1  sample  from  different  population,  EARLY  vocab   § H :1  μ  <  18   § If  this  were  a  non-­‐directional  test,  the  alternative  hypothesis   would  be  μ  ≠  18     Steps:  Milestone  Example   • (1,  2)  Info  we  have:       o Xbar  =  16.5     o S  =  1   o Hypothesized  population  mean  =  μ  =  18;     o YES,  t-­‐test  is  appropriate   • (3)  t  =  (Xbar  –  Xbar     • s Xbar    =    s  /  √N         =  1  /  √24      =    1/4.90      =  0.20  (this  is  the  standard  error)    t    =  (16.5  –  18)  /  0.20   =    -­‐1.5  /  0.20      =        -­‐7.50   • (4)  Degrees  of  freedom  (DF)  =  N-­‐1  =  23   • (5)  From  Appendix  E.6:    need  alpha  &  directionality   o One-­‐tailed  test,  alpha  level  =  .05   o Appendix  row  number  =  critical  value  =  -­‐1.714   o NOTE:  1.714  is  the  t  test  score…  but  we  have  a  one  tailed  test  in  the   negative  direction,  so  we  have  to  change  it  to  -­‐1.714   • (6)  Comparing  our  calculated  t-­‐value  to  critical  value:   o -­‐7.50  <  -­‐1.714,  therefore…  reject  null  hypothesis  because  it  falls  in  the   rejection  region     Summary  View  of  Two  Tests     • The  commonality  between  Z-­‐test  and  one  sample  t-­‐test  is  that  we  are  only   dealing  with  one  set  of  data     Paired  t-­‐test:  Overview   • We  use  this  when  we  have  “paired”  or  related/dependent  observations  (it  is   also  called  “dependent  samples  t-­‐test”)   o We  are  no  longer  dealing  with  just  one  sample  of  data   • The  sets  of  data  may  be  from  the  same  person  at  two  points  in  time,  or  from   people  that  are  related  to  each  other     Paired  t-­‐test:  Assumptions   • Our  data  depends  on  each  other  so  we  do  not  need  to  have  the  assumption  of   independent  sampling   • With  paired  t-­‐test,  we  are  analyzing  different  scores   • Paired  t-­‐test  assumes  distribution  of  the  difference  scores  is  approximately   normal     Difference  Scores   • X  is  the  first  score  of  a  pair  and  X  is  the  second  score  then:   1 2 o D  =  (X1  -2‐  X )  =  difference  score   • Once  we  transform  our  two  sets  of  related  scores  into  one  set  of  difference   scores,  we  can  proceed  with  exactly  the  same  steps  as  with  the  one-­‐sample  t-­‐ test   • D  will  be  negative  if  the  pre  test  is  smaller  than  post  test,  and  vice  versa  


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