Thursday September 26 2013 Paired samples ttest Last

Thursday, September 26, 2013 Paired samples t-test

Last Time • Independent Samples t-test – Numerator: (MA-MB)-(μA- μB) – (μA- μB) always equals 0 – Denominator: (estimated standard error of the difference between two independent means) • Formula for denominator: • Formula for pooled variance: • Questions about these items before we move on? • Next topic: Paired/Related Samples t-tests

Practice problem You are conducting an experiment about methods for teaching reading. You have access to a sample of 10 3 rd graders. You randomly assign half of the group to an experimental reading intervention, and the other half receives instruction as usual in their classrooms. After the intervention, you measure the number of words each child can read correctly in one minute, and obtain the following results. • Group 1 (experimental) scores: 30, 35, 40, 20, 32 • Group 2 scores: 25, 30, 20, 18 • Conduct a t-test to find out whether the groups are different in reading ability at the end of the study. • Show all four steps of hypothesis testing. • Use the t-table from the textbook or the one from

Effect Size for the t Test for Independent Means • Estimated effect size after a completed study • Percentage of the variance in the DV explained by the IV

Power for the t Test for Independent Means (. 05 significance level)

Approximate Sample Size Needed for 80% Power (. 05 significance level)

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Statistical analysis follows design • The one-sample z-test can be used when: – 1 sample – One score per subject – Population mean (μ) and standard deviation (s) are known

Statistical analysis follows design • The one-sample t-test can be used when: – 1 sample – One score per subject – Population mean (μ) is known – but standard deviation (s) is NOT known

Statistical analysis follows design • The independent samples t-test can be used when: – 2 samples – Samples are independent

t Test for Dependent Means • Unknown population mean and variance • Two situations – One sample, two scores for each person • Repeated measures design – Two samples, but pairs of individuals in the samples are related in some way (data points can be grouped into pairs) • Same procedure as t test for single sample, except – Use difference scores – Assume that the population mean is 0

Statistical analysis follows design • The related-samples ttest can be used when: – 1 sample – Two scores per subject

Statistical analysis follows design • The related-samples ttest can be used when: – 1 sample – Two scores per subject - OR - – 2 samples – Scores are related

Performing your statistical test • Difference scores – For each pair of scores, subtract one score from the other – Carry out hypothesis testing with the difference scores • Population of difference scores with a mean of 0 What are all of these “D’s” referring to? Mean of the differences Estimated standard error of the mean of the differences Number of difference scores Test statistic Diff. Expected by chance

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Performing your statistical test (Pre-test) - (Post-test) What are all of these “D’s” referring to? Difference Person Pre-test Post-test scores 1 2 3 4 45 55 40 60 43 49 35 51 2 6 5 9 22 H 0: There is no difference between pre-test and posttest μD = 0 HA: There is a difference between pre-test and posttest μD ≠ 0

Performing your statistical test (Pre-test) - (Post-test) Difference Person Pre-test Post-test scores 1 2 3 4 45 55 40 60 43 49 35 51 2 6 5 9 22 = 5. 5 What are all of these “D’s” referring to?

Performing your statistical test What are all of these “D’s” referring to? Difference Person Pre-test Post-testscores 1 2 3 4 45 55 40 60 43 49 35 51 2 6 5 9 22 MD = 5. 5

Performing your statistical test What are all of these “D’s” referring to? Difference Person Pre-test Post-test scores D - MD (D - MD)2 1 2 3 4 45 55 40 60 43 49 35 51 2 - 5. 5 = 6 - 5. 5 = 5 - 5. 5 = 9 - 5. 5 = 22 MD = 5. 5 -3. 5 12. 25 0. 25 -0. 5 0. 25 3. 5 12. 25 25 = SSD

Performing your statistical test What are all of these “D’s” referring to? Difference Person Pre-test Post-test scores D - MD (D - MD)2 1 2 3 4 45 55 40 60 43 49 35 51 2 6 5 9 22 MD = 5. 5 -3. 5 12. 25 0. 25 -0. 5 0. 25 3. 5 12. 25 25 = SSD

Performing your statistical test What are all of these “D’s” referring to? Difference Person Pre-test Post-test scores D - MD (D - MD)2 1 2 3 4 45 55 40 60 43 49 35 51 2 6 5 9 22 MD= 5. 5 -3. 5 12. 25 0. 25 -0. 5 0. 25 3. 5 12. 25 25 = SSD 2. 9 = s. D

Performing your statistical test What are all of these “D’s” referring to? Difference Person Pre-test Post-test scores D - MD (D - MD)2 1 2 3 4 45 55 40 60 43 49 35 51 2 6 5 9 22 MD = 5. 5 -3. 5 12. 25 0. 25 -0. 5 0. 25 3. 5 12. 25 25 = SSD 2. 9 = s. D ? Think back to the null hypotheses

Performing your statistical test What are all of these “D’s” referring to? Difference Person Pre-test Post-test scores D - MD (D - MD)2 1 2 3 4 45 55 40 60 43 49 35 51 2 6 5 9 22 MD = 5. 5 -3. 5 12. 25 0. 25 -0. 5 0. 25 3. 5 12. 25 25 = SSD 2. 9 = s. D H 0: Memory performance at the post-test are equal to memory performance at the pre-test.

Performing your statistical test What are all of these “D’s” referring to? Difference Person Pre-test Post-test scores D - MD (D - MD)2 1 2 3 4 45 55 40 60 43 49 35 51 2 6 5 9 22 MD= 5. 5 -3. 5 12. 25 0. 25 -0. 5 0. 25 3. 5 12. 25 25 = SSD 2. 9 = s. D This is our tobs

Performing your statistical test What are all of these “D’s” referring to? Difference Person Pre-test Post-test scores D - MD (D - D)2 1 2 3 4 45 55 40 60 43 49 35 51 2 6 5 9 22 MD= 5. 5 -3. 5 12. 25 0. 25 -0. 5 0. 25 a= 0. 05 Two-tailed 3. 5 12. 25 25 = SSD 2. 9 = s. D tobs tcrit

Performing your statistical test What are all of these “D’s” referring to? Difference Person Pre-test Post-test scores D - MD (D - MD)2 1 2 3 4 45 55 40 60 43 49 35 51 2 6 5 9 22 MD = 5. 5 -3. 5 12. 25 0. 25 tobs -0. 5 0. 25 a= 0. 05 Two-tailed tcrit 3. 5 12. 25 25 = SSD tobs=3. 8 2. 9 = s. D - Reject H 0 +3. 18 = tcrit

Performing your statistical test What are all of these “D’s” referring to? Difference Person Pre-test Post-test scores D - MD (D - MD)2 1 2 3 4 45 55 40 60 43 49 35 51 2 6 5 9 22 MD = 5. 5 -3. 5 12. 25 0. 25 tobs -0. 5 0. 25 a= 0. 05 Two-tailed tcrit 3. 5 12. 25 25 = SSD Tobs > tcrit so 2. 9 = s. D reject the H 0 we

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Statistical Tests Summary Design Statistical test (Estimated) Standard error Degrees of freedom One sample, σ known One sample, σ unknown Two independent samples, σ unknown Two related samples (or one sample with repeated measures), σ unknown n– 1 (n. A – 1) + (n. B -1)

Effect Sizes & Power for t Test for Dependent Means Remember we don’t know these Estimated

Approximate Sample Size Needed for 80% Power (. 05 significance level) • Using Power and effect sizes to determine how many participants you need

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Using spss to conduct t-tests • One-sample t-test: Analyze =>Compare Means =>One sample t-test. Select the variable you want to analyze, and type in the expected mean based on your null hypothesis. • Independent samples t-test: Analyze =>Compare Means =>Independent samples t-test. Specify test variable and grouping variable, and click on define groups to specify how grouping variable will identify groups. • Paired or related samples t-test: Analyze =>Compare Means =>Paired samples t-test. Select the variables you want to compare and drag them into the “pair 1”

Practice problem Use SPSS to conduct a paired samples t -test comparing opinions about two different kinds of chocolate chip cookies. Use SPSS to conduct an independent samples t-test comparing opinions about two different samples of one brand.

Using excel to compute ttests • =ttest(array 1, array 2, tails, type) • Select the arrays that you want to compare, specify number of tails (1 or 2) and type of t-test (1=dependent, 2=independent w/equal variance assumed, 3=independent w/unequal variance assumed). • Returns the p-value associated with the t-test.

Next Homework (due Monday, October 1) Chapter 10: 1 -4, 6, 21, 22 Chapter 11: 1, 5, 9, 10, 18, 20