# The t-test. What We Will Cover in This Section. A Research Situation

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1 The -es 1//008 P331 -ess 1 Wha We Will Cover in This Secion Inroducion One-sample -es. Power and effec size. Independen samples -es. Dependen samples -es. Key learning poins. 1//008 P331 -ess A Research Siuaion A high school wans o know if a special SAT preparaion program has helped sudens raise heir scores. They go scores of a group of 5 sudens. Hisorically he mean verbal score for all of heir graduaing seniors is µ 485, bu hey don have he sandard deviaion. The sample has a mean SAT score of 497 wih a sandard deviaion of Wha is he research hypohesis?. Wha is H o? 3. Wha is H a? 4. Wha is he saisical hypohesis? 5. Is his a one-ailed or wo-ailed es? 1//008 P331 -ess 3

2 Draw The Picure (DTP) µ 485, σ? 497 Ŝ 10 1//008 P331 -ess 4 z-es and he Single Sample -es Known saisics : F Ŝ N z-es * Single sample -es No 1//008 P331 -ess 5 Comparing he Formulas Sandard Error of he Mean Tes of he Difference Beween Means z- es σ σ N µ Z σ Single Sample - es ^ S σ N µ ( N 1) σ 1//008 P331 -ess 6

3 Degrees of Freedom (df) Developed from he noion ha when you know ha a group of N numbers sum o S, and if you know N-1 of he numbers, he N h number is fixed. Example. If a group of 4 numbers add up o 15 and hree of he numbers are 5, 6, and, wha is he fourh number? In his case you have N-1 degrees of freedom. 1//008 P331 -ess 7 Back o he Example: Compuaion A high school wans o know if a special SAT preparaion program has program has helped sudens raise heir scores. They go scores from a group of 5 paricipans. Hisorically he mean verbal score for all of heir graduaing seniors is µ 485, bu hey don have he sandard deviaion. The sample has a mean SAT score of 497 wih a sandard deviaion of σ 5 σ ( N 1).00 (4) //008 P331 -ess 8 1//008 P331 -ess 9

4 How o Express he -es #1 (4) 6.00, p<.05 Degrees of Freedom Calculaed value Alpha Level Inerpreaion: ( wih 4 degrees of freedom equals 6.00.) (The probabiliy of geing his resul by chance is less han 5%.) Therefore I rejec he null hypohesis and conclude ha he alernaive hypohesis is rue. 1//008 P331 -ess 10 How o Express he -es # (4) 1.0, n.s. Degrees of Freedom Calculaed value No Significan Inerpreaion: ( wih 4 degrees of freedom equals 1.0.) (The difference beween he means is probably due o chance.) Therefore I fail o rejec he null hypohesis and conclude ha my research hypohesis is wrong. 1//008 P331 -ess 11 Assumpions of Single Sample -es 1. The populaion mean is available.. The populaion disribuion is normal. 3. The observaions are independen. 4. Measuremen is done on an inerval or raio scale. 5. You have he sample Mean Sandard Deviaion (Ŝ) Sample Size (N) 1//008 P331 -ess 1

5 Power and Effec Size 1//008 P331 -ess 13 Power Can he es deec a reamen difference when he difference exiss? POWER is he probabiliy ha he es will correcly rejec a false null hypohesis. A weak saisical es will raise he probabiliy of making a Type II error 1//008 P331 -ess 14 Things Tha Influence Power 1. Alpha level.. One vs. wo-ailed es. 3. Sample size. 1//008 P331 -ess 15

6 Effec Size The magniude or influence of he independen variable on he dependen variable. 1//008 P331 -ess 16 Weaker Effec 1 1//008 P331 -ess 17 Srong Effec 1 1//008 P331 -ess 18

7 Power and Effec Size 1. A powerful (sensiive) saisical es will deec a weak effec.. A weak es will fail o deec a small effec (Type II error). 1//008 P331 -ess 19 Saisical vs. Pracical Significance Large sample sizes increase he power of a es, make i more sensiive. Powerful ess deec relaively small effecs. Small effecs are no necessarily useful (pracical). 1//008 P331 -ess 0 ea ea + df Inerpreed in erms of he amoun of variabiliy accouned for in he dependen variable when one knows he level of he independen variable. Soccer Sudy ea ea ea.40 1//008 P331 -ess 1

8 IMPORTANT! Compue effec size ONLY when you rejec he null hypohesis. Why? Rejecing Ho means your reamen had is expeced effec. Failing o rejec Ho means your reamen was no effecive. I does no make sense o look a he effec size of somehing ha does no work! 1//008 P331 -ess YOU DO NOT COMPUTE EFFECT SIZE WHEN THE RESULTS ARE NOT STATISTICALLY SIGNIFICANT!!!!!! 1//008 P331 -ess 3 Pracice Problems 1//008 P331 -ess 4

9 Memory Booser Problem Sara Bellum is planning o sell a memory booser, a concocion of herbs and minerals inended o improve memory performance. Sara inends o es he effeciveness wih a sample of 16 people by having hem ake he mixure daily for six days. A he end of his ime hey ake a sandardized memory es. The sample scores were M6, s8. For he populaion, µ 0. Is his a one or woailed es? Wha does he picure look like? Wha is he saisical hypohesis? Wha are he degrees of freedom. Wha is he criical value of? 1//008 P331 -ess 5 Sep 1, Draw he Picure µ 0 6 1//008 P331 -ess 6 Sep, Compue he Sandard Error of he Mean S ^ S N 8 S //008 P331 -ess 7

10 Sep 3. Compue - µ (N-1) S 6-0 (15) //008 P331 -ess 8 Sep 4. Decide if is Significan Does exceed he criical value? If yes rejec H o. If no fail o rejec H o. How do your express his resul for publicaion? 1//008 P331 -ess 9 Sep 5. Calculae he Effec Size ea + df ea ea 4.00 ea.375 Wha does his number mean? 1//008 P331 -ess 30

11 Assignmen Homework #10 1//008 P331 -ess 31 Pracice Problems 1//008 P331 -ess 3 Harassmen Seminars Hanz Zoff, well known sexual harassmen counselor waned o evaluae he impac of one long harassmen workshop agains many shorer ones. He did no know which would be beer. Hanz used employees a wo locaions. There were 15 employees in each group. The mean number of complains for he single session was 7, SS 100. For he muliple session group he mean was 68, SS 110. Did workshop frequency have an impac of harassmen? 1. Wha saisical es should Hanz use?. Should his be a one or wo-ailed es? 3. Wha are he degrees of freedom? 4. Wha is H o? 5. Wha is he criical value of a p<.05? 6. Wha is he criical value a p<.01? 1//008 P331 -ess 33

12 Sep 1. Draw he Picure Long Many shor 1//008 P331 -ess 34 Sep. Compue he Pooled Variance s SS + SS 1 p df1+ df 1//008 P331 -ess 35 Sep 3. Compue he Sandard Error S S S + 1 N N P P 1 1//008 P331 -ess 36

13 Sep 4. Compue ( N1+ N ) S 1 1 1//008 P331 -ess 37 Sep 5. Decide if is Significan Sep 6. Wha Conclusion do You Reach? 1//008 P331 -ess 38 Sep 7. Compue ea ea + df 1//008 P331 -ess 39

14 Sudy Problem Fiz Mabu was ineresed in wheher sudens learned beer in comforable chairs versus uncomforable chairs. Fiz did no know which would lead o beer grades. Fiz evaluaed he level of learning under boh he comforable and uncomforable condiions. There were sudens in each group. The mean score for he comforable group was 81, SS 31. For he uncomforable group he mean was 85, SS Wha is he research hypohesis?. Is his a one ailed or woailed es? 3. Wha is he saisical hypohesis? 4. Wha are he degrees of freedom? 5. Wha is he criical value of? 1//008 P331 -ess 40 Solving he Comfor Problem 1. Pooled variance.. Sandard error of he difference. 3. Calculae. 1//008 P331 -ess 41 Comfor Problem, Decisions 1. Wha saisical decision should you reach?. Wha is your conclusion? 3. Should you compue effec size? 4. Wha is he effec size? 5. How do you inerpre he effec size? 1//008 P331 -ess 4

15 Assignmen Homework #11 1//008 P331 -ess 43 1//008 P331 -ess 44

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