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Comparing Two Means Dependent and Independent T-Tests Class 14

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Logic of Inferential Stats Detective Althype: “Tony 'Trout Eyes' Nullhype was at the murder scene.” Tony “Trout Eyes” Nullhype: “No fuggin way! I was at duh church rummage sale!” Dataville Witness Reports Witness 1: Saw Tony at scene Witness 2: Saw Tony at scene Witness 3: Not sure Dataville Witness Reports Witness 1: Saw Tony at scene Witness 2: Saw Tony at scene Witness 3: Not sure Witness 4: Not sure Witness 5: Not sure Witness 6: Not sure Witness 7: Not sure Error

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Logic of Inferential Stats Degree of Certainty All Observations 2 witnesses ID’d Tony = 0.66 confirmation rate 3 witnesses total 2 witness ID’d Tony = 0.29 confirmation rate 7 witnesses total

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Generating Anxiety—Photos vs. Reality: Within Subjects and Between Subjects Designs Problem Statement: Are people as aroused by photos of threatening things as by the physical presence of threatening things? Hypothesis: Physical presence will arouse more anxiety than pictures. Expt’l Hypothesis: Seeing a real tarantula will arouse more anxiety than will spider photos.

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Spider Photos

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WUNDT!!!!

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WITHIN SUBJECTS DESIGN 1.All subjects see both spider pictures and real tarantula 2.Counter-balanced the order of presentation. Why? 3.DV: Anxiety after picture and after real tarantula Data (from spiderRM.sav) Subject Picture (anx. score) Real T (anx. score) 13040 23535 34550 --------- 125039

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Results: Anxiety Due to Pictures vs. Real Tarantula Do the means LOOK different? Are they SIGNIFICANTLY DIFFERENT? Yes Need t-test

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WHY MUST WE LEARN FORMULAS? Don’t computers make stat formulas unnecessary 1. SPSS conducts most computations, error free 2. In the old days—team of 3-4 work all night to complete stat that SPSS does in.05 seconds. Fundamental formulas explain the logic of stats 1. Gives you more conceptual control over your work 2. Gives you more integrity as a researcher 3. Makes you more comfortable in psych forums

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) + (+ ( X ( 5 ) X (365 X 3 y ) = TODDLER FORMULA Point: Knowing the formula without understanding concepts leads to impoverished understanding.

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Logic of Testing Null Hypothesis Inferential Stats test the null hypothesis ("null hyp.") This means that test is designed to CONFIRM that the null hyp is true. In WITHIN GROUPS t-test (AKA "dependent" t-test) null hyp. is that responses in Cond. A and in Cond. B come from same population of responses. Null hyp.: Cond A and Cond B DON'T differ. In BETWEEN GROUPS t-test (AKA "independent" t-test) null hyp. is that responses from Group A and from Group B DON’T differ. If tests do not confirm the null hyp, then must accept ALT. HYPE. Alt. hyp. within-groups: Cond A differs from Cond B Alt. hyp. between-groupsGroup A differs from Group B

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Null Hyp. and Alt. Hyp in Pictures vs. Reality Study Within groups design : Cond. A (all subjs. see photos), then Cond. B (all subs. see actual tarantula) Null hyp? No differences between seeing photos (Cond A) and seeing real T (Cond B) Anxiety ratings Alt. hyp? There is a difference between seeing photos (Cond A) and seeing real T (Cond B)

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T-Test as Measure of Difference Between Two Means 1. Two data samples—do means of each sample differ significantly? 2. Do samples represent same underlying population (null hyp: small diffs) or two distinct populations (alt. hyp: big diffs)? 3. Compare diff. between sample means to diff. we’d expect if null hyp is true 4. Use Standard Error (SE) to gauge variability btwn means. a. If SE small & null hyp. true, sample diffs should be smaller b. If SE big & null hyp. true, sample diffs. should be larger 5. If sample means differ much more than SE, then either: a. Diff. reflects improbable but true random difference w/n true pop. b. Diff. indicates that samples reflect two distinct true populations. 6. Larger diffs. Between sample means, relative to SE, support alt. hyp. 7. All these points relate to both Dependent and Independent t-tests

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Logic of T-Test observed difference between sample means expected difference between population means (if null hyp. is true) t = − SE of difference between sample means Note: Logic the same for Dependent and Independent t-tests. However, the specific formulas differ.

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Mean Difference Relative to SE (overlap) Small: Null Hyp. Supported Mean Difference Relative to SE (overlap) Large: Alternative Hyp. Supported

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S D : The Standard Error of Differences Between Means Sampling Distribution: The spread of many sample means around a true mean. SE: The average amount that sample means vary around the true mean. SE = Std. Deviation of sample means. Formula for SE : SE = s/√n, when n > 30 If sample N > 30 the sampling distribution should be normal. Mean of sampling distribution = true mean. S D = Average amount Var. 1 mean differs from Var. 2 mean in Sample 1, then in Sample 2, then in Sample 3, ---- then in Sample N Note: S D is differently computed in Between-subs. designs.

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S D : The Standard Error of Differences Between Means TARANTULA PICTURE D MEAN MEAN (T mean – P mean) Study 163 3 Study 253 2 Study 342 2 Study 453 2. Ave. 2.25

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S D : The Standard Error of Differences Between Means TARANT. PICT. D D - D (D-D) 2 Sub. 163 3 -. 75.56 Sub. 253 2.25.07 Sub. 342 2.25.07 Sub. 453 2.25.07 X Tarant = 5 X Pic = 2.75 D = 2.25 Σ (D-D) 2 =.77 S D 2 = Sum (D -D) 2 / N - 1; =.77 / 3 =.26 SD = √SD 2 = √.26 =.51 SE of D = σD = SD / √N =.51 / √4 =.51 / 2 =.255 t = D / SE of D = 2.25 /.255 = 8.823

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Small S D indicates that average difference between pairs of variable means should be large or small, if null hyp true? Small S D will therefore increase or decrease our chance of confirming experimental prediction? Small Increase it. Understanding S D and Experiment Power Power of Experiment: Ability of expt. to detect actual differences.

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Assumptions of Dependent T-Test 1. Samples are normally distributed 2. Data measured at interval level (not ordinal or categorical)

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Conceptual Formula for Dependent Samples T-Test t = D − μ D s D / √N D = Average difference between mean Var. 1 – mean Var. 2. It represents systematic variation, aka experimental effect. μD = Expected difference in true population = 0 It represents random variation, aka the the null effect. s D / √N = Estimated standard error of differences between all potential sample means. It represents the likely random variation between means. = Experimental Effect Random Variation

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Dependent (w/n subs) T-Test SPSS Output t = expt. effect / error t = X / SE t = -7 / 2.83 = -2.473 SE = SD / √n 2.83 = 9.807 / √12 Note: Mean = mean diff pic anx - real anx. = 40 - 47 = - 7

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Independent (between-subjects) t-test 1.Subjects see either spider pictures OR real tarantula 2.Counter-balancing less critical (but still important). Why? 3.DV: Anxiety after picture OR after real tarantula Data (from spiderBG.sav) Subject Condition Anxiety 1130 2235 3145 22250 23160 24239

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Assumptions of Independent T-Test DEPENDENT T-TEST 1. Samples are normally distributed 2. Data measured at least at interval level (not ordinal or categorical) INDEPENDENT T-TESTS ALSO ASSUME 3. Homogeneity of variance 4. Scores are independent (b/c come from diff. people).

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Logic of Independent Samples T-Test (Same as Dependent T-Test) observed difference between sample means expected difference between population means (if null hyp. is true) t = − SE of difference between sample means Note: SE of difference of sample means in independent t test differs from SE in dependent samples t-test

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Conceptual Formula for Independent Samples T-Test t = ( X 1 − X 2 ) − (μ 1 − μ 2 ) Est. of SE (X 1 − X 2 ) = Diffs. btwn. samples It represents systematic variation, aka experimental effect. (μ 1 − μ 2 ) = Expected difference in true populations = 0 It represents random variation, aka the the null effect. Estimated standard error of differences between all potential sample means. It represents the likely random variation between means. = Experimental Effect Random Variation

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Computational Formulas for Independent Samples T-Tests t = X 1 − X 2 2 N1N1 N2N2 () s 1 s 2 2 + √ When N 1 = N 2 t = X 1 − X 2 s p s p 2 + √ 2 n1n1 n2n2 When N 1 ≠ N 2 spsp 2 = (n 1 -1)s 1 + (n 2 -1)s 2 2 2 n 1 + n 2 − 2 Weighted average of each groups SE =

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Independent (between subjects) T-Test SPSS Output t = expt. effect / error t = (X 1 − X 2 ) / SE t = -7 / 4.16 = - 1.68

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Dependent (within subjects) T-Test SPSS Output t = expt. effect / error t = X / SE t = -7 / 2.83 = -2.473 SE = SD / √n 2.83 = 9.807 / √12 Note: Mean = mean diff pic anx - real anx. = 40 - 47 = - 7

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Dependent T-Test is Significant; Independent T-Test Not Significant. A Tale of Two Variances Dependent T-Test Independent T -Test SE = 2.83 SE = 4.16

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