Social Statistics t test This week What is

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Social Statistics: t test

Social Statistics: t test

This week ¥ What is t test ¥ Types of t test ¥ TTEST

This week ¥ What is t test ¥ Types of t test ¥ TTEST function ¥ T-test Tool. Pak 2

Why not z-test ¥ ¥ ¥ 3 In most cases, the z-test requires more

Why not z-test ¥ ¥ ¥ 3 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 ¥ Employed by Guinness Brewery, Dublin, Ireland, from 1899 to 1935.

William S. Gossett ¥ 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”) 4

Types of t test 5

Types of t test 5

Degrees of Freedom and t test ¥ Degrees of freedom describes the number of

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 6

One sample t test ¥ Very similar like z test ¥ Use sample statistics

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 7

An example ¥ ¥ ¥ 8 We show 26 babies the two pictures at

An example ¥ ¥ ¥ 8 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)

Step 1: Hypotheses ¥ Sentence: Null: Babies look at the face stimulus for less

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: 9

Step 2: Determine Critical Region ¥ Population variance is not known, so use sample

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 10

Step 3: Calculate t statistic from sample ¥ Central Limit Theorem μ = 30

Step 3: Calculate t statistic from sample ¥ Central Limit Theorem μ = 30 ¥ s. M=s/ =16/ = 3. 14 ¥ ¥ 11

Step 4: Decision and Conclusion ¥ The tobt=1. 59 does not exceed tcrit=1. 708

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. 12

Independent t test ¥ A research design that uses a separate sample for each

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. 13

t Statistic for Independent. Measures Design ¥ The goal of an independent-measures research study:

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 14

t Statistic for Independent. Measures Design ¥ Null hypothesis: “no change = no effect

t Statistic for Independent. Measures Design ¥ Null hypothesis: “no change = no effect = no difference” ¥ H 0: μ 1 - μ 2 = 0 ¥ Alternative hypothesis: “there is a difference” ¥ 15 H 1: μ 1 - μ 2 ≠ 0

T test formula Value for degrees of freedom: df = df 1 + df

T test formula Value for degrees of freedom: df = df 1 + df 2 16

An Example Group 1 7 3 3 2 3 8 8 5 17 5

An Example Group 1 7 3 3 2 3 8 8 5 17 5 4 6 10 10 5 1 1 4 3 5 7 1 9 2 5 2 12 15 4 Group 2 5 4 4 5 5 7 8 8 9 8 3 2 5 4 4 6 7 7 5 6 4 3 2 7 6 2 8 9 7 6

T test steps ¥ Step 1: A statement of the null and research hypotheses.

T test steps ¥ Step 1: A statement of the null and research hypotheses. ¥ ¥ 18 Null hypothesis: there is no difference between two groups Research hypothesis: there is a difference between the two groups

T test steps ¥ Step 2: setting the level of risk (or the level

T test steps ¥ Step 2: setting the level of risk (or the level of significance or Type I error) associated with the null hypothesis ¥ 19 0. 05

T test steps ¥ Step 3: Selection of the appropriate test statistic Determine which

T test steps ¥ Step 3: Selection of the appropriate test statistic Determine which test statistic is good for your research ¥ Independent t test ¥ 20

T test steps ¥ Step 4: computation of the test statistic value ¥ 21

T test steps ¥ Step 4: computation of the test statistic value ¥ 21 t= 0. 14

T test steps ¥ Step 5: determination of the value needed for the rejection

T test steps ¥ Step 5: determination of the value needed for the rejection of the null hypothesis ¥ 22 T Distribution Critical Values Table

T test steps ¥ Step 5: (cont. ) ¥ Degrees of freedom (df): approximates

T test steps ¥ Step 5: (cont. ) ¥ Degrees of freedom (df): approximates the sample size ¥ ¥ ¥ Two-tailed or one-tailed ¥ ¥ 23 Group 1 sample size -1 + group 2 sample size -1 Our test df = 58 Directed research hypothesis one-tailed Non-directed research hypothesis two-tailed

T test steps ¥ Step 6: A comparison of the obtained value and the

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 ¥ 24

T test steps ¥ Step 7 and 8: make a decision ¥ 25 What

T test steps ¥ Step 7 and 8: make a decision ¥ 25 What is your decision and why?

Interpretation ¥ How to interpret t(58) = 0. 14, p>0. 05, n. s. 26

Interpretation ¥ How to interpret t(58) = 0. 14, p>0. 05, n. s. 26

Excel: T. TEST function ¥ T. TEST (array 1, array 2, tails, type) array

Excel: T. TEST function ¥ T. TEST (array 1, array 2, tails, type) array 1 = the cell address for the first set of data ¥ array 2 = 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 ¥ 27

Excel: T. TEST function ¥ It does not compute the t value ¥ It

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) 28

Excel Tool. Pak ¥ Select t-Test: Two-Sample Assuming Equal Variances t-Test: Two-Sample Assuming Equal

Excel Tool. Pak ¥ 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 29 Variable 1 5. 43333 11. 70229885 30 7. 979885057 0 58 -0. 137103112 0. 44571206 1. 671552763 0. 891424121 2. 001717468 Variable 2 5. 53333 4. 257471264 30

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

Effect size ¥ If two groups are different, how to measure the difference among them ¥ Effect size ES: effect size : the mean for Group 1 : the mean for Group 2 SD: the standard deviation from either group 30

Effect size ¥ A small effect size ranges from 0. 0 ~ 0. 2

Effect size ¥ A small effect size ranges from 0. 0 ~ 0. 2 ¥ ¥ A medium effect size ranges from 0. 2 ~ 0. 5 ¥ ¥ ¥ 31 The two groups are different A large effect size is any value above 0. 50 ¥ ¥ Both groups tend to be very similar and overlap a lot 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%