Hypothesis Testing with t Tests - University of Michigan

1 Hypothesis Testing with Hypothesis Testing with ttTe s t sTe s t sAl Cl kFArlo Clark-FoosUsing Samples to Estimate Using Samples to Estimate Population ParametersPopulation ParametersyAcknowledge errorgySmaller samples, less spread)(2 MX1)( =NMXsThe The ttStatisticStatisticThe The ttStatisticStatisticyIndicates the distance of a sample mean pfrom a population mean in terms of the standard errorss=NsM= Hypothesis Tests : Hypothesis Tests : SingleSingle--Sample Sample ttTe s t sTe s t s Guinness is the best beer available, it does d d l ll ll not need advertising as its quality will sell it, and those who do not drink it are to be sympathized with rather than advertised to sympathized with rather than advertised Tests : Hypothesis Tests : SingleSingle--Sample Sample ttTe s t sTe s t syHypothesis test in which we compare data from one sample to a population for which we know the mean but not the standard of Freedom.

2 The number of scores that are free to vary when estimating a population parameter from a sample df = N 1 (for a Single-Sample t Test)One Tailed vs Two Tailed TestsOne Tailed vs Two Tailed TestsOne Tailed vs. Tw o Ta i l e d Te s t sOne Tailed vs. Tw o Ta i l e d Te s t sSix Steps for Hypothesis TestingSix Steps for Hypothesis TestingSix Steps for Hypothesis TestingSix Steps for Hypothesis Testing1. Identifyy2. State the hypotheses3 Characteristics of the comparison 3. Characteristics of the comparison distribution4 Critical values4. Critical values5. Calculate6. DecideSingleSingle--Sample Sample t t Test: ExampleTest: ExampleSingleSingleSample Sample t t Test: ExampleTest: ExampleyParticipation in therapy sessionsppyyContract to attend 10 sessionsy = 4 6y = :y6, 6, 12, 7, 8 SingleSingle--Sample Sample t t Test: ExampleTest: ExampleSingleSingleSample Sample t t Test: ExampleTest: Pop 1: All clients who sign contract Pop 2: All clients who do not sign contractpg Distribution of means Test: Population mean is known but not Test: Population mean is known but not standard deviation single-sample t testxAssumptionsSingleSingle--Sample Sample t t Test: ExampleTest.

3 ExampleSingleSingleSample Sample t t Test: ExampleTest: the null and research hypothesesyp H0: Clients who sign the contract will attend the same number of sessions as those who do not sign the 1= 2 H1: Clients who sign the contact will attend a different number of sessions than those who do not sign the contractwho do not sign the 1 2 SingleSingle--Sample Sample t t Test: ExampleTest: ExampleSingleSingleSample Sample t t Test: ExampleTest: characteristics of comparison distribution M= = , )(2 MX M , sM= )(= = Sample t t Test: ExampleTest: ExampleSingleSingleSample Sample t t Test: ExampleTest: critical values or cutoffs df = N-1 = 5-1 = 4, p =.

4 05 t = Sample t t Test: ExampleTest: ExampleSingleSingleSample Sample t t Test: ExampleTest: the test ) ()(= = =MsMt 6Mk d a decision > , we reject the null Clients who sign a contract will attend more sessions than those who do not sign a Results in APA FormatReporting Results in APA FormatReporting Results in APA FormatReporting Results in APA the symbol for the test statistic ( , z or t) the degrees of freedom in an equal sign and then the value of the test statistic (2 decimal places) a comma and then whether the pvalue associated with the test statistic was less than or greater than the cutoff pvalue of 05or greater than the cutoff pvalue of.

5 05t(4) 2 87 p < 05t(4) = , p < .05 Hypothesis Tests for two samplesHypothesis Tests for two samplesHypothesis Tests for two samplesHypothesis Tests for two samplesyConditions must be equivalent q(controlled) for a fair testyBeer Tasting: Between-groups vs. Within-groupsPairedPaired--Samples Samples t t Te s tTe s tPairedPairedSamples Samples t t Te s tTe s tyUsed to compare 2 means for a within-pfgroups design, a situation in which every participant is in both samples ppp(paired/dependent)yDifference Scores: X1 Y1, X2 Y2, ..PairedPaired--Samples Samples t t Te s tTe s tPairedPairedSamples Samples t t Te s tTe s null & research characteristics of characteristics of comparison distributionH: = H0.

6 1= 2 PairedPaired--Samples Samples t t Te s tTe s tPairedPairedSamples Samples t t Te s tTe s test statisticM)( 6Mk d iiMMsMt)( = a decisionIndependent Samples Independent Samples ttTe s tTe s tIndependent Samples Independent Samples ttTe s tTe s tyUsed to compare 2 means for a between-pfgroups design, a situation in which each participant is assigned to only one Samples Independent Samples ttTe s tTe s tIndependent Samples Independent Samples ttTe s tTe s tyWhat percentage of cartoons do men and pgwoman consider funny?yWomen: 84 97 58 90yMen:88 90 52 97 86 Independent Samples Independent Samples ttTe s tTe s tIndependent Samples Independent Samples ttTe s tTe s Population 1: Women exposed to humorous cartoonsPopulation 2: Men exposed to humorous cartoons Distribution: Differences between meansxNot mean differences Assumptions are the samexWe meet one here: DV is interval/scaleIndependent Samples Independent Samples ttTe s tTe s tIndependent Samples Independent Samples ttTe s tTe s null and research hypotheses H0: 1= 2 Women will categorize the same number of cartoons as funny as will menfunny as will men.

7 H1: 1 2 Women will categorize a Wo e w catego e a different number of cartoons funny as as will Samples Independent Samples ttTe s tTe s tIndependent Samples Independent Samples ttTe s tTe s characteristics of comparison distribution H0: 1= 2 Measure of spread:a)Calculate variance for each sampleb)Pool variances accounting for sample sizeb)Pool variances, accounting for sample sizec)Convert from squared standard deviation to squared standard errord)Add the two variancese)Take square root to get estimated standard error for distribution of differences between distribution of differences between Samples Independent Samples ttTe s tTe s tIndependent Samples Independent Samples ttTe s tTe s ta)Calculate variance for each sample22() === 141N 22() 1314 4YM 2()

8 === Independent Samples Independent Samples ttTe s tTe s tIndependent Samples Independent Samples ttTe s tTe s tyDegrees of FreedomgdfX= N 1 = 4 1 = 3dfX N1 4 1 3df N1 5 1 4dfY= N 1 = 5 1 = 4dfTo t a l= dfX+ dfY= 3 + 4 = 7 Pooled VariancePooled VariancePooled VariancePooled Varianceb)Pool variances, accounting for sample sizeyWeighted average of the two estimates of variance one from each sample that are calculated when di idpd pl conducting an independent samples t 222 XYPooledXYTotalTotaldfdfsssdfdf =+ 34 =+ +=Independent Samples Independent Samples ttTe s tTe s tIndependent Samples Independent Samples ttTe s tTe s tc)Convert from squared standard deviation to 878s2311 878s)qsquared standard = =d)Add the two += + =d)Add the two variancese)

9 Take square root to get estimated standard error for distribution of differences between =Independent Samples Independent Samples ttTe s tTe s tIndependent Samples Independent Samples ttTe s tTe s critical values dfTo t a l= 7p = .05 t= a test statistic()() ()()()XY XYXYD ifferenceDifferenceMMMMtss ==() 847t == Samples Independent Samples ttTe s tTe s tIndependent Samples Independent Samples ttTe s tTe s a decisionyFail to reject null hypothesisFail to reject null Hypothesis Men and women find cartoons equally humorous