Inferential Stats for Two-Group Designs

Inferential Statistics

• Used to infer conclusions about the population based on data collected from sample

• Do the results occur frequently or rarely by chance?

• Two-group designs: – one control group vs. experimental group– two experimental groups– Two samples representing two populations

Inferential Statistics

• Null hypothesis: differences between groups are due to chance (i.e., not due to the manipulation of IV).– Results are not significant

• Results are Significant: results would occur rarely by chance alone. Therefore, likely due to the manipulation of the IV.– p< .05 or p < .01

Single samples t-test

• Compared single sample with a population– Given population mean– Derived sample mean, standard deviation and standard error of

the mean– Conducted t-test – Determined critical t value (one-tailed vs. two-tailed) based on df

– Reject H0 : the sample comes from a population that is different from the comparison population.

– Sustain H0 : the sample and comparison population come from the same population.

T-test (Difference between Means)

• t-test: Inferential statistical test used to evaluate the difference between two groups or samples.

• Two types: – t-test for independent samples (between-subjects design)– t-test for paired samples (within-subjects design)

T-test for independent samples

• Compares means of 2 different samples of participants.

• Assesses the differences between these 2 samples.

• Do participants in each sample perform so similarly so that they are likely to come from the same population?

• Do participants in each sample perform so differently so that they are likely to represent 2 different populations?

Assumptions of independent samples t-test

• Data are interval-ratio scale• Normally distributed (bell shaped)• Observations of one group are independent of other group. • Homogeneity of variance

– Variance: shows spread the scores are from the mean (standard deviation squared)

– The variance of 2 populations represented by each sample should be equal.

Independent samples t-test

• Two groups of 8 salesclerks, each tested only once.

• IV: dress style of customers• levels:

– Sloppy Clothing customers– Dressy Clothing customers

• DV: response time of clerks to help customers

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Degrees of freedom for independent groups: df = N - 2

Independent samples t-test

• Formula pg 198

• Numerator: difference between 2 sample means• Denominator: standard error of the difference between means of

independent samples.

• Rationale: according to H0, μ1 = μ2, so the difference between population means is zero.

– according to H1, μ1 ≠ μ2, and their difference should be large enough to be significant.

Independent samples t-test

• Rationale continued:

• If we took an infinite number of samples from each population, calculated their means, subtracted the means from each other, and plotted the difference, we would get a distribution of differences between sample means.

• When we take the standard deviation of this new distribution, we get the standard error of the difference between means of independent samples.

Independent samples t-test

• Steps:– 1) calculate means of each sample. – 2) calculate the variance of each sample.– 3) determine the std error of difference between means.– 4) calculate independent samples t-test– 5) determine t critical (based on df = N-2; alpha level, one vs

two-tailed)– 6) make decision: reject or do not reject null hypothesis

Factors that Increase power

• To increase power in independent samples:

– Increase differences between the means (i.e., choose an IV that will produce greater differences between groups)

– Reduce variability of raw scores in each condition (variance will be smaller)

– Increase sample size

Textbook example pg. 197 Group Statistics

condition N Mean Std. Deviation Std. Error

Mean itemsrecalled spaced study 10 22.0000 2.10819 .66667

massed study 10 16.9000 2.51440 .79512

Independent Samples Test

Levene's Test for Equality of Variances t-test for Equality of Means

F Sig. t df Sig. (2-tailed) Mean

Difference Std. Error Difference

95% Confidence Interval of the Difference

Lower Upper itemsrecalled Equal variances

assumed .827 .375 4.915 18 .000 5.10000 1.03763 2.92003 7.27997

Equal variances not assumed 4.915 17.469 .000 5.10000 1.03763 2.91527 7.28473

Hw problem # 1

Independent Samples Test

Levene's Test for Equality of Variances t-test for Equality of Means

F Sig. t df Sig. (2-tailed) Mean

Difference Std. Error Difference

95% Confidence Interval of the Difference

Lower Upper studytime Equal variances

assumed .545 .475 -.784 12 .449 -2.286 2.917 -8.642 4.070

Equal variances not assumed -.784 11.276 .449 -2.286 2.917 -8.687 4.116

Group Statistics

gender N Mean Std. Deviation Std. Error

Mean studytime males 7 19.00 6.110 2.309

females 7 21.29 4.716 1.782

Correlated-groups t-test or paired samples t-test

• One sample where each participant is tested twice (within-subjects design).

• Measures whether there is a difference in the sample means and if this difference is greater than expected by chance.

– H0: there is no difference between the scores of person A in condition 1 and condition 2.

– H1: person A performs differently in condition 1 than condition 2.

• Determine difference scores: difference between participants’ performance in condition 1 and performance in condition 2.

Correlated-samples t-test

• Difference scores: pg 204, table 9.3

• Formula pg. 204

• Numerator: mean of difference scores minus zero• Denominator: standard error of difference scores

– Standard deviation of the difference scores divided by √N

• Degrees of freedom: N - 1

Paired-samples t-test textbook pg. 204

Paired Samples Statistics

Mean N Std. Deviation Std. Error

Mean Pair 1 concretewords 13.2500 8 2.76457 .97742

abstractwords 10.7500 8 1.83225 .64780

Paired Samples Test

Paired Differences

t df Sig. (2-tailed) Mean Std. Deviation Std. Error

Mean

95% Confidence Interval of the Difference

Lower Upper Pair 1 concretewords -

abstractwords 2.50000 1.85164 .65465 .95199 4.04801 3.819 7 .007

Paired samples t-test

• Steps:– 1) calculate the difference scores for each participant.– 2) calculate the mean of difference scores. – 3) determine the std deviation of difference scores. – 4) calculate the std error of the difference scores: divide # 3 by

square root of N– 5) calculate paired samples t-test– 6) determine t critical (based on df = N-1; alpha level, one vs

two-tailed)– 7) make decision: reject or do not reject null hypothesis

Assumptions of paired samples t-test

• Data are interval-ratio scale• Normally distributed (bell shaped)• Observations are correlated; dependent on each other• Homogeneity of variance

– Variance: shows spread the scores are from the mean (standard deviation squared)

– The variance of 2 populations represented by each condition should be equal.

Hw prob. # 3

Paired Samples Statistics

Mean N Std. Deviation Std. Error

Mean Pair 1 beforesports 42.3333 6 2.58199 1.05409

aftersports 43.8333 6 2.56255 1.04616

Paired Samples Test

Paired Differences

t df Sig. (2-tailed) Mean Std. Deviation Std. Error

Mean

95% Confidence Interval of the Difference

Lower Upper Pair 1 beforesports - aftersports -1.50000 .54772 .22361 -2.07480 -.92520 -6.708 5 .001