Comparing Means: t-tests for One Sample & Two Related Samples
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1 Comparing Means: -Tess for One Sample & Two Relaed Samples
2 Using he z-tes: Assumpions -Tess for One Sample & Two Relaed Samples The z-es (of a sample mean agains a populaion mean) is based on he assumpion ha he sample means are normally disribued. For his o be rue, one of he following mus also be rue: The underlying populaion (e.g., of es scores) is normally disribued The sample size (n) is sufficienly large o approximae normaliy by way of he cenral limi heorem Addiionally, we mus know he populaion sandard deviaion (σ)
3 The -Saisic Wha if we don know σ? In mos real-world siuaions in which we wan o es a hypohesis, we do no know he populaion sandard deviaion σ If we don know σ we can compue a es saisic using s, bu his saisic will no longer be normally disribued, so we can no longer use he z es saisic Why? s is variable across samples and is sampling disribuion is no normally disribued s 2 is disribued as a chi-square disribuion, which we ll alk abou near he end of he semeser
4 The Sampling Disribuions of s 2 and s Sample 1 M 74.2 Sample 2 M 73.4 Suden Score Suden Score s s 6.76 s s n 5 Sample 3 M 67.6 Suden Score s s 3.44
5 The -Saisic If we compue somehing like z, bu using s insead of σ, we ge a saisic ha follows he disribuion Remember: Similarly, z M M, M s M, where M where s M n s n
6 Disribuion of he -Saisic -Tess for One Sample & Two Relaed Samples (n = 5)
7 Disribuion of he -Saisic & Sample Size You can hink of he saisic as an "esimaed z-score." The esimaion comes from he fac ha we are using he sample variance o esimae he unknown populaion variance. The value of degrees of freedom, df = n - 1, deermines how well he disribuion of approximaes a normal disribuion and how well he saisic represens a z-score. For large samples (large df), he esimaion is very good and he saisic will be very similar o a z-score. For small samples (small df), he saisic will provide a relaively poor esimae of z. For large df, he disribuion will be nearly normal, bu for small df, he disribuion will be flaer and more spread ou han a normal disribuion.
8 Disribuion of he -Saisic -Tess for One Sample & Two Relaed Samples The shape of he -disribuion depends on he number of degrees of freedom
9 Disribuion of he -Saisic -Tess for One Sample & Two Relaed Samples Smaller samples (wih fewer df) require greaer values o rejec H 0 (4), α = 0.05 (200), α = 0.05
10 -Disribuion Table One-ailed es α/2 - Two-ailed es α α/2 Level of significance for one-ailed es Level of significance for wo-ailed es df
11 The One-Sample -Tes: Example Research Hypohesis H 1 : µ Dr.M µ AVG Null Hypohesis H 0 : µ Dr.M = µ AVG 70.0? We sample 5 (i.e., n=5) sudens from Dr. M s class, adminiser he es and find ha heir average score is 75, wih a sample sandard deviaion of 7.0. Do we reain or rejec he null hypohesis? Assume a wo-ailed es, wih α = 0.05
12 The One-Sample -Tes: Seps -Tess for One Sample & Two Relaed Samples 1. Use disribuion able o find criical -value(s) represening rejecion region (denoed by cri or α ) 2. Compue -saisic For daa in which I give you raw scores, you will have o compue he sample mean and sample sandard deviaion 3. Make a decision: does he -saisic for your sample fall ino he rejecion region?
13 -Disribuion Table One-ailed es α/2 - Two-ailed es α α/2 Level of significance for one-ailed es Level of significance for wo-ailed es df
14 -Disribuion Table One-ailed es α/2 - Two-ailed es α α/2 Level of significance for one-ailed es Level of significance for wo-ailed es df
15 Compue -Saisic -Tess for One Sample & Two Relaed Samples 70.0 s 7.0 n 5 M df M s n
16 Compue -Saisic -Tess for One Sample & Two Relaed Samples 70.0 s 7.0 n 5 M df M s n n
17 criical region (exreme 5%) rejec H 0 middle 95% reain H 0 (4), α= 0.05, 2-ailed es M µ from H = 1.60 reain H 0 : no difference
18 The One-Sample -Tes: A Full Example Moon illusion example: how large mus moon be a zenih o appear equivalen in size o moon a horizon? x = size zenih /size horizon Null Hypohesis H 0 : µ =? Research Hypohesis H 1 : µ? x = {1.73,1.06,2.03, 1.40, 0.95,1.13,1.41,1.73,1.63,1.56} Do we accep or rejec he null hypohesis? Assume a wo-ailed es, wih α = 0.05
19 The One-Sample -Tes: A Full Example Moon illusion example: how large mus moon be a zenih o appear equivalen in size o moon a horizon? x = size zenih /size horizon Null Hypohesis H 0 : µ = 1 Research Hypohesis H 1 : µ 1 x = {1.73,1.06,2.03, 1.40, 0.95,1.13,1.41,1.73,1.63,1.56} Do we reain or rejec he null hypohesis? Assume a wo-ailed es, wih α = 0.05
20 -Disribuion Table One-ailed es α/2 - Two-ailed es α α/2 Level of significance for one-ailed es Level of significance for wo-ailed es df
21 -Disribuion Table One-ailed es α/2 - Two-ailed es α α/2 Level of significance for one-ailed es Level of significance for wo-ailed es df
22 1. Compue sample mean and SD 2. Use hese values o compue -saisic X X sum df M s n M x i n 2 x 2 SS x n s SS n 1
23 1. Compue sample mean and SD 2. Use hese values o compue -saisic X X sum x i M n 10 x SS x n 10 SS s n 1 9 (Remember, µ=1 under H 0 ) df n M s n ; rejec H 0
24 Hypohesis Tesing wih he -Saisic Boh he sample size and he sample variance influence he oucome of a hypohesis es. The sample size is inversely relaed o he esimaed sandard error. Therefore, a large sample size increases he likelihood of a significan es (for an exising effec). The sample variance, on he oher hand, is direcly relaed o he esimaed sandard error. Therefore, a large variance decreases he likelihood of a significan es (for an exising effec).
25 -Tess for Two Relaed Samples: Repeaed Measures The relaed-samples -es allows researchers o evaluae he mean difference beween wo reamen condiions using he daa from a single sample. This es can also be called he repeaed-measures -es, he machedsamples -es, or he paired-samples -es In a repeaed-measures design, a single group of individuals is obained and each individual is measured in boh of he reamen condiions being compared. Thus, he daa consis of wo scores for each individual.
26 -Tess for Two Relaed Samples: Mached Subjecs The relaed-samples es can also be used for a similar design, called a mached-subjecs design, in which each individual in one reamen is mached one-o-one wih a corresponding individual in he second reamen. The maching is accomplished by selecing pairs of subjecs so ha he wo subjecs in each pair have idenical (or nearly idenical) scores on he variable ha is being used for maching.
27 -Saisic for Two Relaed Samples The repeaed-measures saisic allows researchers o es a hypohesis abou he populaion mean difference beween wo reamen condiions using sample daa from a repeaedmeasures research sudy. In his siuaion i is possible o compue a difference score for each individual: difference score = D = x 2 x 1 where x 1 is he person s score in he firs reamen and x 2 is he score in he second reamen.
28 Two Relaed Samples: Example Family herapy for anorexic girls x 1 : weigh before reamen x 2 : weigh afer reamen X 1 X 2 D = X 2 X
29 -Saisic for Two Relaed Samples The sample of difference scores is used o es hypoheses abou he populaion of difference scores. The null hypohesis saes ha he populaion of difference scores has a mean of zero: H : 0 0 D 2 1 The alernaive hypohesis saes ha here is a sysemaic difference beween reamens ha causes he difference scores o be consisenly posiive (or negaive) and produces a non-zero mean difference beween he reamens: H : 0 1 D
30 -Saisic for Two Relaed Samples This should seem very familiar: he repeaed-measures saisic forms a raio wih exacly he same srucure as he one-sample saisic. M D D D s s df n 1 M D D D M n D The numeraor of he saisic measures he difference beween he sample mean and he hypohesized populaion mean. The only differences are ha: he sample mean and sandard deviaion are compued for he difference scores D The populaion mean under H 0 is always 0 The number of degrees of freedom (df) is compued based on he difference scores
31 The Repeaed-Measures -Tes: Full Example Does family herapy affec he weigh gained by anorexic girls? x 1 : weigh before reamen x 2 : weigh afer reamen D = x 2 x 1 Null Hypohesis H 0 : µ D = 0 Research Hypohesis H 1 : µ D 0 Do we reain or rejec he null hypohesis? Assume a wo-ailed es, wih α = 0.05
32 -Disribuion Table One-ailed es α/2 - Two-ailed es α α/2 Level of significance for one-ailed es Level of significance for wo-ailed es df
33 1. Compue sample mean and SD 2. Use hese values o compue -saisic X 1 X 2 D = X 2 X 1 D Sum M D D i n 10 D SS D n 10 SS s n 1 9 (Remember, µ D =0 under H 0 ) df n M D s D n D ; rejec H 0
34 Measuring Effec Size for Mean Differences Because he significance of a reamen effec is deermined parially by he size of he effec and parially by he size of he sample, you canno assume ha a significan effec is also a large effec. Therefore, a measure of effec size is usually compued along wih he hypohesis es. Cohen s d: d 1 0 Cohen s d measures he size of he reamen effec in erms of he sandard deviaion.
35 Measuring Effec Size for Mean Differences Of course we usually do no have all of he populaion parameers. Therefore, we usually compue an esimae of he effec size. For z-ess: dˆ M 0 For one-sample -ess: ˆd M 0 s For relaed-samples -ess: ˆ D D 0 d M s D
36 Measuring Effec Size for he -Saisic For he moon illusion example: 1.0 s n 10 M Cohen s d: dˆ M s
37 Measuring Effec Size for he -Saisic For he moon illusion example: 1.0 s n 10 M Cohen s d: ˆ M d 1.36 s 0.341
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