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### This week

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

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### Why not z-test

   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”

### William S. Gossett

4    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”)

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### Degrees of Freedom and t test

 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

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### One sample t test

 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

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### An example

     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)

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### Step 1: Hypotheses

 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:

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### Step 2: Determine Critical Region

 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

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### Step 3: Calculate t statistic from sample

 Central Limit Theorem  μ = 30  s M =s/ =16/ = 3.14

n

26 

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### Step 4: Decision and Conclusion

 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.

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### Independent t test

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

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### t Statistic for Independent Measures Design

 The goal of an independent-measures research study:  To evaluate the difference of the means between two populations.

 Mean of first population: μ 1  Mean of second population: μ 2  Difference between the means: μ 1 μ 2

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### t Statistic for Independent Measures Design

 Null hypothesis: “no change = no effect = no difference”  H0: μ 1 μ 2 = 0  Alternative hypothesis: “there is a difference”  H1: μ 1 μ 2 ≠ 0

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### T test formula

t

x

1 

x

2   (

n

1  1 )

n

1 2

s

1  

n

2 (

n

 2 2  1 )

s

2 2   

n

1

n

1

n

2

n

2   𝑥 1 : is the mean for Group 1 𝑥 2 : is the mean for Group 2 𝑛 1 : is the number of participants in Group 1 𝑛 2 : is the number of participants in Group 2 𝑠 1 2 : is the variance for Group 1 𝑠 2 2 : is the variance for Group 20 Value for degrees of freedom: df = df1 + df2

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### An Example

Group 1 7 3 3 2 3 8 8 5 8 5 5 4 6 10 10 5 1 1 4 3 2 12 15 4 5 7 1 9 2 5 Group 2 5 4 4 5 5 7 8 8 9 8 7 7 5 6 3 2 5 4 4 6 8 9 7 6 4 3 2 7 6 2

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### T test steps

 Step 1: A statement of the null and research hypotheses.

 Null hypothesis: there is no difference between two groups

H

0 :  1   2  Research hypothesis: there is a difference between the two groups

H

1 :  1   2

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### T test steps

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

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### T test steps

 Step 3: Selection of the appropriate test statistic   Determine which test statistic is good for your research Independent t test

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### T test steps

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

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### T test steps

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

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### T test steps

 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

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### T test steps

 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

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### T test steps

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

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### Interpretation

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

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### Excel: T.TEST function

 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

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### Excel: T.TEST function

 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 ToolPak

 Select

t-Test: Two-Sample Assuming Equal Variances

t-Test: Two-Sample Assuming Equal Variances Mean Variance Observations Pooled Variance Hypothesized Mean Difference df t Stat P(T<=t) one-tail t Critical one-tail P(T<=t) two-tail t Critical two-tail

Variable 1

5.433333333

11.70229885

30 7.979885057

0 58 -0.137103112

0.44571206

1.671552763

0.891424121

2.001717468

Variable 2

5.533333333

4.257471264

30 29

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### Effect size

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

ES

X

1 

X

2

SD

ES: effect size

X

1

X

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

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### Effect size

   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=0  the two groups have no difference and overlap entirely ES=1  the two groups overlap about 45%