Social Statistics: t test

A repeated-measures study (a.k.a dependent study) is one in which a single sample of individuals is measured more than once on the same dependent variable.

Main benefit: two sets of data are from the same subjects.

T test for dependent

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Three professors at University of Alabama studied the effects of resource and regular classrooms on the reading achievement of learning-disabled children. A group of children was test before they take the 1-year daily instruction and after they took the 1-year daily instruction. Which statistical test we should use?

Example

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1

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n

DDn

Dt

T test for dependent

: the sum of all the difference between groups : the sum of the differences squared between groupsn : the number of pairs of observations

D

2D

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Pre-test Post-test D D2

3 7 4 165 8 3 94 6 2 46 7 1 15 8 3 95 9 4 164 6 2 45 6 1 13 7 4 166 8 2 47 8 1 18 7 -1 17 9 2 46 10 4 167 9 2 48 9 1 18 8 0 09 8 -1 19 4 -5 258 4 -4 167 5 -2 47 6 -1 16 9 3 97 8 1 18 12 4 16

T test for dependent

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Step1: a statement of the null and research hypotheses

T test for dependent

pretestposttestH :0

pretestposttestH :1

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Step2: setting the level of risk (or the level of significance or Type I error) associated with null hypothesis 0.05

T test for dependent

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Step3: selection of the appropriate test statistics T test for dependent = t test for paired

samples = t test for correlated samples

T test for dependent

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Step4: computation of the test statistic value t=2.45

T test for dependent

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Step5: determination of the value needed for rejection of the null hypothesis T Distribution Critical Values Table df=n-1=25-1=24 One tailed: because research hypothesis

is directed

T test for dependent

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Step6: a comparison of the t value and the critical value 2.45>1.711 Reject the null hypothesis

T test for dependent

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Step7 and 8: time for a decision There is the difference between pre-test

and post-test: the post-test scores are higher than the pre-test scores.

T test for dependent

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T.TEST (array1, array2, tails, type) array1 = the cell address for the first set

of data array2 = the cell address for the second

set of data tails: 1 = one-tailed, 2 = two-tailed type: 1 = a paired t test; 2 = a two-

sample test (independent with equal variances); 3 = a two-sample test with unequal variances

Excel: T.TEST function

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It does not computer the t value It returns the likelihood that the resulting t

value is due to chance Less than 1% of the possibility that two tests

are different due to chance the two tests are difference due to other reasons than chance.

Excel T.TEST()

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t-Test: Paired Two Sample for Means option

Excel ToolPak

t-Test: Paired Two Sample for Means

  pretest posttestMean 6.32 7.52Variance 2.976666667 3.343333333Observations 25 25Pearson Correlation 0.050718341Hypothesized Mean Difference 0df 24t Stat -2.449489743P(T<=t) one-tail 0.010991498t Critical one-tail 1.710882067P(T<=t) two-tail 0.021982997t Critical two-tail 2.063898547

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Repeated-measures design reduces or limits the variance, by eliminating the individual differences between samples.

Advantages of theRepeated-Samples Design

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Carryover effect (specifically associated with repeated-measures design): subject’s score in second measurement is altered by a lingering aftereffect from the first measurement.

Problems With theRepeated-Samples Design

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Types of t test

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A researcher is interested in a new technique to improve SAT verbal scores. It is known that SAT verbal scores have μ=500 σ=100.

She randomly selects n=30 students from this population, and has them undergo her training technique. Students are given analogy questions, and are shocked each time they get an answer wrong.

The sample then writes the SAT, and gets M = 560.

Example I (select appropriate methods and make the test, the same as

followings)

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A social psychologist is interested in whether people feel more or less hopeful following a devastating flood in a small rural community. He randomly selects n=10 people and asks them to report how hopeful the feel using a 7-point scale from extremely hopeful (1) to neutral (4) to extremely unhopeful (7)

The researcher is interested in whether the responses are consistently above or below the midpoint (4) on the scale, but has no hypothesis about what direction they are likely to go.

His sample reports M=4.7, s = 1.89.

Example II

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To test the hypothesis that people give out more candy to kids in cute costumes than scary ones, I hire 20 kids to work for me. Ten are randomly assigned to wear cute bunny costumes, and the other ten wear Darth Vader costumes.

I drop the kids off in random parts of the city, and count the total pieces of candy each has after 1 hour of trick-or-treat.

Cute bunnies: M = 120, s = 10 Darth Vaders: M = 112, s = 12

Example III

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We are testing the effects of moderate amounts of alcohol on driving performance. We make the hypothesis that even a small amount of beer will degrade driving performance (an increase in obstacles hit).

To test our hypothesis, we have n=5 subjects drive around a course on Big Wheels™ covered with cardboard cutouts of children and furry animals, and we record the number of cutouts they hit. Then, they drink one beer, and do the course again; again we record the number of cutouts hit.

What is a potential confound with this experiment?

Example IV

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We want to determine if IU LIS faculty publish more than the national average of 4 papers per year (per person). We take a random sample of n=12 IU LIS professors and survey the number of papers each has published, obtaining M=6.3, s=1.13.

Example V

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I want to know which dog is responsible for the holes in my yard. I buy 10 German Shepherds, 10 Beagles, and randomly assign each dog to its own yard. At the end of the day, the Beagles have dug M=11.3 holes, s=2.1, and the Shepherds have dug M=5.4 holes, s=1.9. Test my hypothesis that Beagles dig more holes than German Shepherds.

Example VI

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We want to know if noise affects surgery performance. We randomly select a sample of 9 surgeons, and have them perform a hand-eye coordination task (not while performing surgery, of course). The surgeons first perform the task in a quiet condition, and then we have them perform the same task under a noisy condition. Test the hypothesis that noise will cause poorer performance on the task.

Example VII

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ETS reports that GRE quantitative scores for people who have not taken a training course are μ=555, σ=139. We take a sample of 10 people from this population and give them a new preparation course. Test the hypothesis that their test scores differ from the population.

Example VIII

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