Social Statistics: t test

What is t test Types of t test TTEST function T-test ToolPak

This week

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In most cases, the z-test requires more information than we have available

We do inferential statistics to learn about the unknown population but, ironically, we need to know characteristics of the population to make inferences about it

Enter the t-test: “estimate what you don’t know”

Why not z-test

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Employed by Guinness Brewery, Dublin, Ireland, from 1899 to 1935.

Developed t-test around 1905, for dealing with small samples in brewing quality control.

Published in 1908 under pseudonym “Student” (“Student’s t-test”)

William S. Gossett

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Types of t test

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Degrees of freedom describes the number of scores in a sample that are free to vary.

degrees of freedom = df = n-1 The larger, the better

Degrees of Freedom and t test

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Very similar like z test Use sample statistics instead of

population parameters (mean and standard deviation)

Evaluate the result through t test table instead of z test table

One sample t test

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We show 26 babies the two pictures at the same time (one with his/her mother, the other a scenery picture) for 60 seconds, and measure how long they look at the facial configuration.

Our null assumption is that they will not look at it for longer than half the time, μ = 30

Our alternate hypothesis is that they will look at the face stimulus longer and face recognition is hardwired in their brain, not learned (directional)

Our sample of n = 26 babies looks at the face stimulus for M = 35 seconds, s = 16 seconds

Test our hypotheses (α = .05, one-tailed)

An example

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Sentence: Null: Babies look at the face stimulus for

less than or equal to half the time Alternate: Babies look at the face

stimulus for more than half the time Code Symbols:

Step 1: Hypotheses

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Population variance is not known, so use sample variance to estimate

n = 26 babies; df = n-1 = 25 Look up values for t at the limits of

the critical region from our critical values of t table

Set α = .05; one-tailed tcrit = +1.708

Step 2: Determine Critical Region

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Central Limit Theorem μ = 30 sM=s/ =16/ = 3.14

Step 3: Calculate t statistic from sample

n 26

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The tobt=1.59 does not exceed tcrit=1.708

∴ We must retain the null hypothesis Conclusion: Babies do not look at the

face stimulus more often than chance, t(25) = +1.59, n.s., one-tailed. Our results do not support the hypothesis that face processing is innate.

Step 4: Decision and Conclusion

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A research design that uses a separate sample for each treatment condition is called an independent-measures (or between-subjects) research design.

Independent t test

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The goal of an independent-measures research study: To evaluate the difference of the means

between two populations. Mean of first population: μ1 Mean of second population: μ2 Difference between the means: μ1-

μ2

t Statistic for Independent-Measures Design

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Null hypothesis: “no change = no effect = no difference” H0: μ1- μ2 = 0

Alternative hypothesis: “there is a difference” H1: μ1- μ2 ≠ 0

t Statistic for Independent-Measures Design

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222

211

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2)1()1(

nnnn

nnsnsn

xxt

T test formula

: is the mean for Group 1: is the mean for Group 2: is the number of participants in Group 1: is the number of participants in Group 2: is the variance for Group 1: is the variance for Group 20

Value for degrees of freedom: df = df1 + df2

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 Group 1       Group 2 7 5 5    5 3 43 4 7    4 2 33 6 1    4 5 22 10 9    5 4 73 10 2    5 4 68 5 5    7 6 28 1 2    8 7 85 1 12    8 7 98 4 15    9 5 75 3 4    8 6 6

An Example

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Step 1: A statement of the null and research hypotheses. Null hypothesis: there is no difference

between two groups

Research hypothesis: there is a difference between the two groups

T test steps

210 : H

211 : H

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Step 2: setting the level of risk (or the level of significance or Type I error) associated with the null hypothesis 0.05

T test steps

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Step 3: Selection of the appropriate test statistic Determine which test statistic is good

for your research Independent t test

T test steps

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Step 4: computation of the test statistic value t= 0.14

T test steps

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Step 5: determination of the value needed for the rejection of the null hypothesis T Distribution Critical Values Table

T test steps

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Step 5: (cont.)

Degrees of freedom (df): approximates the sample size

Group 1 sample size -1 + group 2 sample size -1 Our test df = 58

Two-tailed or one-tailed Directed research hypothesis one-tailed Non-directed research hypothesis two-tailed

T test steps

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Step 6: A comparison of the obtained value and the critical value 0.14 and 2.001 If the obtained value > the critical value,

reject the null hypothesis If the obtained value < the critical value,

retain the null hypothesis

T test steps

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Step 7 and 8: make a decision What is your decision and why?

T test steps

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How to interpret t(58) = 0.14, p>0.05, n.s.

Interpretation

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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 compute the t value It returns the likelihood that the

resulting t value is due to chance (the possibility of the difference of two groups is due to chance)

Excel: T.TEST function

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Select t-Test: Two-Sample Assuming Equal Variances

Excel ToolPak

t-Test: Two-Sample Assuming Equal Variances

  Variable 1 Variable 2Mean 5.433333333 5.533333333Variance 11.70229885 4.257471264Observations 30 30Pooled Variance 7.979885057Hypothesized Mean Difference 0df 58t Stat -0.137103112P(T<=t) one-tail 0.44571206t Critical one-tail 1.671552763P(T<=t) two-tail 0.891424121t Critical two-tail 2.001717468 

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If two groups are different, how to measure the difference among them Effect size

Effect size

SD

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ES: effect size : the mean for Group 1 : the mean for Group 2SD: the standard deviation from either group

1X

2X

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A small effect size ranges from 0.0 ~ 0.2 Both groups tend to be very similar and

overlap a lot A medium effect size ranges from 0.2 ~

0.5 The two groups are different

A large effect size is any value above 0.50 The two groups are quite different

ES=0the two groups have no difference and overlap entirely

ES=1the two groups overlap about 45%

Effect size

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