New User Special Price Expires in

Let's log you in.

Sign in with Facebook


Don't have a StudySoup account? Create one here!


Create a StudySoup account

Be part of our community, it's free to join!

Sign up with Facebook


Create your account
By creating an account you agree to StudySoup's terms and conditions and privacy policy

Already have a StudySoup account? Login here

PSYS 054, Statistics for Psychological Science 2/23/16

by: Delaney Row

PSYS 054, Statistics for Psychological Science 2/23/16 PSYS 054

Marketplace > University of Vermont > Psychlogy > PSYS 054 > PSYS 054 Statistics for Psychological Science 2 23 16
Delaney Row
GPA 3.65

Preview These Notes for FREE

Get a free preview of these Notes, just enter your email below.

Unlock Preview
Unlock Preview

Preview these materials now for free

Why put in your email? Get access to more of this material and other relevant free materials for your school

View Preview

About this Document

These notes cover paired t-tests and independent samples t-test
Statistics for Psychological Science
Keith Burt
Class Notes
PSYS 054, Psych stats, Statistics for Psychological Science
25 ?




Popular in Statistics for Psychological Science

Popular in Psychlogy

This 7 page Class Notes was uploaded by Delaney Row on Friday February 26, 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 36 views. For similar materials see Statistics for Psychological Science in Psychlogy at University of Vermont.


Reviews for PSYS 054, Statistics for Psychological Science 2/23/16


Report this Material


What is Karma?


Karma is the currency of StudySoup.

You can buy or earn more Karma at anytime and redeem it for class notes, study guides, flashcards, and more!

Date Created: 02/26/16
PSYS  054  –  Statistics  For  Psychological  Science   Notes  for  the  week  of  2/22/16  –  2/26/16     2/23/16     Paired  t-­‐test:  Overview   • A  crucial  point  to  remember  is  that  we  are  using  paired  t-­‐tests  when  we  are   using  a  pair  of  “related”  scores   • Examples:   o Repeated  measures  on  the  same  person  (having  a  pre-­‐test  and  a  post-­‐ test)   o Ratings  from  pairs  of  spouses  or  family  members   o Longitudinal  data     Paired  t-­‐test:  Assumptions   • Paired  t-­‐test  loses  the  assumption  of  independence  of  observations   • With  the  paired  t-­‐test,  we  are  observing  difference  scores   o This  is:  one  score  (#1)  minus  the  other  (#2)  (also  can  be  thought  of   pre-­‐test  minus  post-­‐test)   • Paired  t-­‐test  assumed  that  the  distribution  of  the  difference  scores  is   relatively  normal     Difference  Scores   • Notation:  with  the  pre-­‐test  of  a  pair  being  Xsub1  and  the  post-­‐test  being   Xsub2…   o Then,  D  =  (Xsub1  –  Xsub2)  =  the  difference  score   • Once  we  get  the  difference  score,  we  are  basically  following  all  the  steps  we   do  in  a  one-­‐sample  t-­‐test,  with  the  difference  score  being  the  main  data  we   are  looking  at   o Degrees  of  freedom  for  paired  t-­‐tests  will  always  be:  (Number  of   difference  scores  -­‐1)   o Use  the  M  and  SD  of  difference  scores  in  the  formula     Paired  t-­‐test:  Hypothesis  specification     Symbol   Interpretation       µ 1   Population  mean  of  score  set  #1         µ   Population  mean  of  score  set  #2     2     μ   Population  mean  of  difference  scores   D     • In  all  cases… Dμ  1  µ2  -­‐  µ   • For  a  two  tailed  test…   o H : 0   D  =  0   § This  is  always  going  to  be  the  null  hypothesis  for  any  paired  t-­‐ test  (it’s  saying  that  there  is  no  difference  in  scores)   o H : 1   D  ≠  0   o Example  from  class:   § “A  stats  professor  is  interested  in  whether  or  not  students’   scores  on  Exam  #2  will  be  higher  or  lower  than  scores  on  Exam   #1.  The  professor  has  no  specific  prediction  about  which  exam   scores  will  be  higher  or  lower.”   • For  a  one  tailed  test  in  the  positive  direction…   o H : 0   D  =  0   § Notice  the  null  hypothesis  stays  the  same   o H : 1   D  >  0   § For  a  one  tailed  test,  the  alternative  hypothesis  must  have  the   mean  difference  either  be  great  or  less  than  zero,  depending  on   its  direction   o Example  from  class:   § “35  individuals  diagnosed  with  Major  Depressive  Disorder   (“clinical  depression”)  are  given  a  baseline  questionnaire  of   depressive  symptoms  at  the  first  visit  of  a  clinical  trial  for  a   new  medication.    They  complete  the  questionnaire  again  after   taking  the  medication  for  6  weeks.  The  researchers   hypothesize  that  the  medication  will  reduce  depressive   symptoms  over  time.”   • For  a  one  tailed  test  in  the  negative  direction…   o H : 0   D  =  0   o H : 1   D  <  0   o Example  from  class:   § “A  researcher  is  interested  in  studying  the  effect  of  age  on   perceived  conflict  in  romantic  relationships.    She  gives  each   partner  from  15  couples  a  questionnaire  assessing  their   reports  of  conflict  in  the  relationship.    The  scores  are  set  up  so   that  younger  partners  are  set  #1  and  older  partners  are  set  #2.     The  researcher  has  reason  to  predict  that  older  partners  will   report  more  conflict.”     Paired  t-­‐test:  Formula  and  Notation     • What  you  need:   o Mean  of  difference  scores  (Dbar)   o SD  of  difference  scores  (S  sub  D)   o Standard  error  of  difference  scores  (SDbar)   o Number  of  difference  scores  (N  sub  D)     Paired  t-­‐test:  Example   • Baby  sign  language  research     Paired  t-­‐test:  Sample  Data   • Sample  of  8  infants  assessed  two  times  (at  15  months  and  24  months)   o 15  months  =  x1  (or  pre-­‐test)  and  24  months  =  x2  (pre-­‐test)   • Researchers  are  predicting  an  increase  over  time,  so  it  will  be  a  one  tailed   negative  test   o Since  it’s  x1-­‐x2  and  x2  is  suspected  to  be  bigger,  it  will  be  one  tailed   negative     Paired  t-­‐test:  Calculations   • Hypotheses   o H : 0  μD  =  0   o H : 1   D μ  <  0   • Calculations   o D 1 =  6-­‐9  = 2  -­‐3;  D  =  2-­‐7  =  -­‐5…   •   2   D   Dbar D  –  (D  –  Dbar) Baby 15 24   Dbar ID Months months D     -­‐3 -­‐2.75 -­‐0.25 0.06 1 6 9 -3   -­‐5 -­‐2.75 -­‐2.25 5.06 2 2 7 -5     0 -­‐2.75 2.75 7.56 3 5 5 0     -­‐7 -­‐2.75 -­‐4.25 18.06 4 1 8 -7   5 4 8 -4 -­‐4 -­‐2.75 -­‐1.25 1.56   6 8 10 -2   -­‐2 -­‐2.75 0.75 0.56   7 9 6 3   3 -­‐2.75 5.75 33.06   8 3 7 -4 -­‐4 -­‐2.75 -­‐1.25 1.56   2   -   ∑(D-­‐Dbar) Mean of difference scores = 2.75   =    67.48 SD of difference scores = 3.10 • D  for  the  most  part  is  negative  (what  researchers  expected)   • We  are  replacing  the  two  sets  of  scores  with  the  difference  score     o We  are  finding  M  and  SD  of  difference  scores,  not  of  the  other  sets  of   scores     Paired  t-­‐test:  Calculations  (continued)   • Dbar  =  mean  of  difference  scores  =  -­‐2.75   • s D    SD  of  difference  scores  =  3.10   • Paired  t-­‐test  formula:   o t  =  Dbar  / D (s  /D  √N )   o    =  -­‐2.75  /  (3.10  /  √8)   o    =  -­‐2.75  /  (3.10  /  2.83)   o    =  -­‐2.75  /  1.10   o    =  -­‐2.50    (df  =  8-­‐1  =  7)   • Degrees  of  freedom  =  7   • Note:  the  denominator  in  the  formula  is  already  accounting  for  standard   error     Paired  t-­‐test  Example:  Decision   • t  =  -­‐2.50   • One-­‐tailed  test,  negative  direction   o (α  =  .05)   o df  =  8-­‐1  =  7   • From  Appendix  E.6,  critical  value  =  -­‐1.895   • -­‐2.50  <  -­‐1.895,  therefore:    reject  the  null  hypothesis     Reporting  t-­‐test:  APA  Style   • Italicize  notation  letters,  but  not  numbers   • Include  degrees  of  freedom  in  parentheses   • If  calculated  t  falls  in  rejection  region,  write  “p  <  .05”  or    report  exact  p  if  you   are  using  SPSS   o Or  write  “n.s.”  (meaning  ‘not  significant’),  e.g.  “  t(11)  =  0.53,  n.s.”   • So  for  our  paired  t-­‐test  example:   o t(7)  =  -­‐2.50,  p  <  .05   o NOTE:  lower  case  t  should  be  italicized  and  then  the  7  in  parentheses     Independent  Samples  t-­‐test   • The  most  common  t-­‐test  used  in  actual  research   o We  will  most  likely  be  using  this  test  on  our  final  projects!   • This  test  compares  independent  sample  means     o For  example,  the  difference  between  mean  of  variables  X  an1  X , 2 when  these  represent  scores  from  different,  unrelated  individuals   One-­‐sample  t-­‐test Paired  t-­‐test Independent  samples  t -­‐test Compares  ONE  sample   Compares  TWO  sample  means   Compares  TWO  sample  means,   mean  to  a  hypothesized   that  are  related  (through   even  if  they  are  unrelated population  mean difference  scores) Independent  Samples  t:  Hypothesis   • In  the  population,  which  group  has  the  larger  mean?   • Similarly  to  a  paired  t-­‐test,  hypotheses  are  written  in  the  form  of  population   mean  differences   o H : 0  1µ 2-­‐  µ  =  0    (null  hypothesis)   § This  is  saying  that  the  mean  scores  from  each  group  will  be   equal/have  no  difference   o H : 1  1µ 2-­‐  µ  ≠  0    (two-­‐tailed  research  hypothesis)   o H : 1  1µ 2-­‐  µ  <  0    (one-­‐tailed  research  hypothesis)  …OR…   o H :    µ  -­‐  µ  >  0    (one-­‐tailed  research  hypothesis)   1 1 2 • t  is  calculated  in  a  similar  fashion  as  well,  BUT:   • The  standard  error  calculations  are  somewhat  more  complex   • Degrees  of  freedom  are  calculated  differently     Real-­‐Life  Research  Using  Independent  Sample  t-­‐test   • Examples  from  class:   o Fingerson  (2008),  studying  psychopathy  and  trauma:  “…an   independent-­‐samples  t-­‐test  found  no  significant  differences  between   psychopaths  and  non-­‐psychopaths  with  regard  to  their  exposure  to   trauma…  ”   o Gibson  (2008),  studying  ethnicity  (Caucasian  vs.  African-­‐American)   and  exposure  to  thin-­‐ideal  images  in  female  college  athletes:  “…an   independent  samples  t-­‐test  was  conducted…  the  test  was  significant;   white  female  athletes  have  higher  exposure  to  magazines  than  non-­‐ white  female  athletes…  t(79)  =  -­‐2.44,  p  <  .01…”     Formula  for  Independent  Samples  t     • The  denominator  represents  another  type  of  standard  error;  the  variances  of   the  group   • Degrees  of  freedom  is  different  here:  df 1=(N 2  +  N  -­‐  2)   • NOTE:  memorize  the  different  degrees  of  freedom  formulas;  they  will  not  be   on  the  formula  sheet  for  exams!     Independent  Samples  t-­‐test:  Assumptions   • Independent  t-­‐test  assumes:   o (1)  Data  are  independently  sampled…   o (2)  From  a  normally  distributed  population…   § Normality  assumption   o (3)  Variance  of  the  two  groups  is  the  same  in  the  population   § “Homogeneity  of  variance  assumption”   Independent  Samples  t-­‐test:  Example   • Researchers  want  to  know  if  learning  baby  signs  impacts  language   development  (positively  or  negatively)  by  age  2   o This  will  be  a  two  tailed  test  because  there  is  no  specific  direction  the   researchers  are  going  in   • Group  A:    asked  to  pay  special  attention  to  baby’s  language   • Group  B:    taught  a  series  of  baby  signs   • Outcome:  standardized  language  assessment  score  (1-­‐5  range)   • N  =  10  infants  per  group   • Results:   o Group  A:  M  =  3.20,  SD  =  1.79   o Group  B:  M  =  4.00,  SD  =  2.61     Thinking  About  Populations   • μ 1  =  mean  of  whatever  population  Group  A  is  drawn  from   • μ 2  =  mean  of  whatever  population  Group  B  is  drawn  from   • We  want  to  know:  could  samples  have  come  from  populations  with  the  same   mean?   o If  true,  difference  in  group  means  is  due  only  to  sampling  error   • H :0  μ1  2­‐  μ  =  0   • H :1  μ1  2    μ ≠  0   o This  alternative  hypothesis  shows  it’s  a  two  tailed  test     Independent  t-­‐test:  Applying  Steps   1. Info:    M  and  SD  from  two  independent  samples   2. Appropriate  test:  independent  samples  t-­‐test!   2 2 3. t  = 1X bar  -2‐  X bar  /s1r(s /1  +2  s2/N )   4. df  =  (1  +2  N  -­‐  2)  =  10  +  10  -­‐  2  =  18   5. Two-­‐tailed  test,  .05  alpha  level,  degrees  of  freedom:  use  Appendix  E.6   6. Compare  calculated  t  statistic  to  critical  values   • Note:  a  two-­‐tailed  test  has  two  critical  values,  positive  and  negative!   • If  calculated  t  is  greater  than  positive  or  less  than  negative  critical  value,   reject  the  null  hypothesis,  otherwise  retain     Calculating  t  for  the  Example   • This  can  be  tricky;  we  are  given  SD  but  we  need  variance,  so  we  need  to   square  each  SD  to  get  the  variances  of  each  group   • t  =  (3.20  -­‐  4.00)  /  sqr(3.20/10  +  6.81/10)     =  (-­‐.80)  /  sqr(.320  +  .681)     =  -­‐.8  /  sqr(1.001)     =  -­‐.8  /  1.00     =  -­‐.8   • Or  since  t’s  are  written  with  one  or  more  numbers  before  the  decimal  and   two  after,  t(18)  =  -­‐0.80.     Appendix  E.6  for  Example:  Two-­‐Tailed   • Critical  values  of  t  for  a  two-­‐tailed  test,  alpha  of  .05,  df=18,  are:    ±  2.101   • -­‐0.80  is  not  more  extreme  than  -­‐2.10,  therefore:  we  will  retain  the  null   hypothesis   o There  is  not  enough  information  to  suggest  that  these  two  samples   come  from  different  populations…      


Buy Material

Are you sure you want to buy this material for

25 Karma

Buy Material

BOOM! Enjoy Your Free Notes!

We've added these Notes to your profile, click here to view them now.


You're already Subscribed!

Looks like you've already subscribed to StudySoup, you won't need to purchase another subscription to get this material. To access this material simply click 'View Full Document'

Why people love StudySoup

Bentley McCaw University of Florida

"I was shooting for a perfect 4.0 GPA this semester. Having StudySoup as a study aid was critical to helping me achieve my goal...and I nailed it!"

Allison Fischer University of Alabama

"I signed up to be an Elite Notetaker with 2 of my sorority sisters this semester. We just posted our notes weekly and were each making over $600 per month. I LOVE StudySoup!"

Bentley McCaw University of Florida

"I was shooting for a perfect 4.0 GPA this semester. Having StudySoup as a study aid was critical to helping me achieve my goal...and I nailed it!"


"Their 'Elite Notetakers' are making over $1,200/month in sales by creating high quality content that helps their classmates in a time of need."

Become an Elite Notetaker and start selling your notes online!

Refund Policy


All subscriptions to StudySoup are paid in full at the time of subscribing. To change your credit card information or to cancel your subscription, go to "Edit Settings". All credit card information will be available there. If you should decide to cancel your subscription, it will continue to be valid until the next payment period, as all payments for the current period were made in advance. For special circumstances, please email


StudySoup has more than 1 million course-specific study resources to help students study smarter. If you’re having trouble finding what you’re looking for, our customer support team can help you find what you need! Feel free to contact them here:

Recurring Subscriptions: If you have canceled your recurring subscription on the day of renewal and have not downloaded any documents, you may request a refund by submitting an email to

Satisfaction Guarantee: If you’re not satisfied with your subscription, you can contact us for further help. Contact must be made within 3 business days of your subscription purchase and your refund request will be subject for review.

Please Note: Refunds can never be provided more than 30 days after the initial purchase date regardless of your activity on the site.