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Comparing Two Means: One-sample & Paired-sample t-tests Lesson 13 Inferential Statistics Hypothesis testing Drawing conclusions about differences between groups Are differences likely due to chance? Comparing means t-test: 2 means Analysis of variance: 2 or more means ~ Comparing 2 means: t-tests One-sample t-test Is sample likely from particular population? Paired-Sample t-test 2 dependent (related) samples Independent-samples t-test 2 unrelated samples ~ The One-sample t-test Evaluating hypothesis about population taking a single sample Does it likely come from population? Test statistics z test if s known t test if s unknown ~ t statistic X t sX df n 1 Example: One-sample t-test Survey: college students study 21 hr/wk Do Coe students study 21 hrs/week? Select sample (n = 16) s unknown Nondirectional hypothesis: H0 : = 21; H1 : 21 reject H0 if increase or decrease PASW/SPSS: Test value = 21 Assumed from H0 ~ PASW One Sample T Test Menu Analyze Compare Means One-Sample T Test Dialog box Test Variable(s) (DV) Test Value (value of testing against) Options (to change confidence intervals) ~ PASW Output *1-tailed probability: divide Sig. 2-tailed by 2 Paired-Samples t-tests 2 samples are statistically related Less affected by individual differences reduces variance due to error Repeated-measures 2 measurements on same individual Matched-subjects Match pairs on some variable(s) Split pairs into 2 groups ~ Difference Scores Find difference between each score D = X2 - X1 Requires n1 scores equal n2 scores Calculate mean D D D N And standard deviation of D 2 ~ DD sD n 1 Repeated-measures 2 measurements of same individual Pretest-posttest design measure each individual twice pretest treatment posttest compare scores ~ Matched-subjects Match individuals on important characteristic individuals that are related IQ, GPA, married, etc Assign to different treatment groups each group receives different levels of independent variable ~ Assumptions: Related Samples Population of difference scores is normal Observations within each treatment independent scores for each subject in a group is independent of other subjects scores ~ Related-samples Hypotheses Nondirectional H 0: D = 0 H 1: D 0 Directional H 0: D > 0 H 1: D < 0 Remember: it depends on the direction of the prediction ~ Sample Statistics Mean difference D D n Mean for single sample X X n Standard Deviation: Related-samples Single sample df D N 1 D D df N 1 sD n 1 X X 2 2 s n 1 Estimated Standard Error Calculate same as single sample use standard deviation of difference scores sD sD N Test Statistic Related-samples t test tobs D D sD Since D= 0 tobs D sD Example Is arachnophobia limited to real spiders or is a picture enough? Participants 12 spider phobic individuals Manipulation (IV) Each person exposed to a real spider & picture of same spider at two different times Outcome (DV): Anxiety PASW Paired-Sample T Test Data entry 1 column each DV Menu Analyze Compare Means Paired-Sample T Test Dialog box Paired Variable(s) (DV) Options (to change confidence intervals) ~ PASW Output Reporting the Results On average, participants experienced significantly greater anxiety to real spiders (M = 47.00, SE = 3.18) than to pictures of spiders (M = 40.00, SE = 2.68), t(11) = −2.47, p < .05