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Using t-tests Basic introduction and 1-sample t-tests Statistics for the Social Sciences Psychology 340 Spring 2010

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PSY 340 Statistics for the Social Sciences Outline (for content up to Exam 2) Review t-tests (note: next exam is March 4, right before spring break) –One sample, related samples, independent samples –Effect sizes with t-tests –Confidence intervals in t-test type designs

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PSY 340 Statistics for the Social Sciences 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 (σ)are known

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PSY 340 Statistics for the Social Sciences 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 Population standard deviation (σ) is NOT known

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PSY 340 Statistics for the Social Sciences Testing Hypotheses –Step 1: State your hypotheses –Step 2: Set your decision criteria –Step 3: Collect your data –Step 4: Compute your test statistics Compute your estimated standard error Compute your t-statistic Compute your degrees of freedom –Step 5: Make a decision about your null hypothesis Hypothesis testing: a five step program

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PSY 340 Statistics for the Social Sciences Performing your statistical test What are we doing when we test the hypotheses? –Consider a variation of our memory experiment example Population of memory patients MemoryTest μ is known Memory treatment Memory patients Memory Test X Compare these two means Conclusions: the memory treatment sample are the same as those in the population of memory patients. they aren’t the same as those in the population of memory patients H0:H0: HA:HA:

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PSY 340 Statistics for the Social Sciences Performing your statistical test What are we doing when we test the hypotheses? Real world (‘truth’) H 0 : is false (is a treatment effect) Two populations XAXA they aren’t the same as those in the population of memory patients H 0 : is true (no treatment effect) One population XAXA the memory treatment sample are the same as those in the population of memory patients.

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PSY 340 Statistics for the Social Sciences Performing your statistical test What are we doing when we test the hypotheses? –Computing a test statistic: Generic test Could be difference between a sample and a population, or between different samples Based on standard error or an estimate of the standard error

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PSY 340 Statistics for the Social Sciences Performing your statistical test Test statistic One sample z One sample t identical

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PSY 340 Statistics for the Social Sciences Performing your statistical test Test statistic Diff. Expected by chance Standard error One sample z One sample t different don’t know this, so need to estimate it

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PSY 340 Statistics for the Social Sciences Performing your statistical test Test statistic Diff. Expected by chance Standard error Estimated standard error One sample z One sample t different “Unbiased estimate of the population standard deviation” This is the same as the “sample standard deviation”

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PSY 340 Statistics for the Social Sciences Performing your statistical test Test statistic Diff. Expected by chance Standard error Estimated standard error One sample z One sample t different don’t know this, so need to estimate it Degrees of freedom

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PSY 340 Statistics for the Social Sciences One sample t-test The t-statistic distribution (a transformation of the distribution of sample means transformed) –Varies in shape according to the degrees of freedom New table: the t-tablet-table

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PSY 340 Statistics for the Social Sciences One sample t-test If test statistic is here Fail to reject H 0 Distribution of the t-statistic If test statistic is here Reject H 0 –To reject the H 0, you want a computed test statistics that is large The alpha level gives us the decision criterion New table: the t-tablet-table The t-statistic distribution (a transformation of the distribution of sample means transformed)

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PSY 340 Statistics for the Social Sciences One sample t-test New table: the t-table One tailed - or - Two-tailed Critical values of t t crit Degrees of freedom df α - level

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PSY 340 Statistics for the Social Sciences One sample t-test What is the t crit for a two-tailed hypothesis test with a sample size of n = 6 and an α-level of 0.05? Distribution of the t-statistic α = 0.05 Two-tailed n = 6 df = n - 1 = 5 t crit = + 2.571

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PSY 340 Statistics for the Social Sciences One sample t-test Distribution of the t-statistic α = 0.05 One-tailed n = 6 df = n - 1 = 5 t crit = +2.015 What is the t crit for a one-tailed hypothesis test with a sample size of n = 6 and an α-level of 0.05?

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PSY 340 Statistics for the Social Sciences One sample t-test An example: One sample t-test Memory experiment example: We give a n = 16 memory patients a memory improvement treatment. How do they compare to the general population of memory patients who have a distribution of memory errors that is Normal, μ = 60? After the treatment they have an average score of = 55, s = 8 memory errors. Don’t know σ Do know s

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PSY 340 Statistics for the Social Sciences One sample t-test An example: One sample t-test Memory experiment example: We give a n = 16 memory patients a memory improvement treatment. How do they compare to the general population of memory patients who have a distribution of memory errors that is Normal, μ = 60? After the treatment they have an average score of = 55, s = 8 memory errors. Step 1: State your hypotheses H0:H0: the memory treatment sample are the same (or worse) as those in the population of memory patients. HA:HA: they perform better than those in the population of memory patients μ Treatment > μ pop = 60 μ Treatment < μ pop = 60

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PSY 340 Statistics for the Social Sciences One sample t-test Memory experiment example: We give a n = 16 memory patients a memory improvement treatment. Step 2: Set your decision criteria H 0 : μ Treatment > μ pop = 60 H A : μ Treatment < μ pop = 60 α = 0.05 One -tailed How do they compare to the general population of memory patients who have a distribution of memory errors that is Normal, μ = 60? After the treatment they have an average score of = 55, s = 8 memory errors. An example: One sample t-test

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PSY 340 Statistics for the Social Sciences One sample t-test An example: One sample t-test Memory experiment example: We give a n = 16 memory patients a memory improvement treatment. Step 2: Set your decision criteria α = 0.05 One -tailed How do they compare to the general population of memory patients who have a distribution of memory errors that is Normal, μ = 60? After the treatment they have an average score of = 55, s = 8 memory errors. H 0 : μ Treatment > μ pop = 60 H A : μ Treatment < μ pop = 60

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PSY 340 Statistics for the Social Sciences One sample t-test An example: One sample t-test Memory experiment example: We give a n = 16 memory patients a memory improvement treatment. α = 0.05 One -tailed Step 3: Collect your data How do they compare to the general population of memory patients who have a distribution of memory errors that is Normal, μ = 60? After the treatment they have an average score of = 55, s = 8 memory errors. H 0 : μ Treatment > μ pop = 60 H A : μ Treatment < μ pop = 60

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PSY 340 Statistics for the Social Sciences One sample t-test An example: One sample t-test Memory experiment example: We give a n = 16 memory patients a memory improvement treatment. α = 0.05 One -tailed Step 4: Compute your test statistics = -2.5 How do they compare to the general population of memory patients who have a distribution of memory errors that is Normal, μ = 60? After the treatment they have an average score of = 55, s = 8 memory errors. H 0 : μ Treatment > μ pop = 60 H A : μ Treatment < μ pop = 60

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PSY 340 Statistics for the Social Sciences One sample t-test An example: One sample t-test Memory experiment example: We give a n = 16 memory patients a memory improvement treatment. α = 0.05 One -tailed Step 4: Compute your test statistics t = -2.5 How do they compare to the general population of memory patients who have a distribution of memory errors that is Normal, μ = 60? After the treatment they have an average score of = 55, s = 8 memory errors. H 0 : μ Treatment > μ pop = 60 H A : μ Treatment < μ pop = 60

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PSY 340 Statistics for the Social Sciences One sample t-test An example: One sample t-test Memory experiment example: We give a n = 16 memory patients a memory improvement treatment. α = 0.05 One -tailed Step 5: Make a decision about your null hypothesis How do they compare to the general population of memory patients who have a distribution of memory errors that is Normal, μ = 60? After the treatment they have an average score of = 55, s = 8 memory errors. t crit = -1.753 H 0 : μ Treatment > μ pop = 60 H A : μ Treatment < μ pop = 60

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PSY 340 Statistics for the Social Sciences One sample t-test An example: One sample t-test Memory experiment example: We give a n = 16 memory patients a memory improvement treatment. α = 0.05 One -tailed Step 5: Make a decision about your null hypothesis How do they compare to the general population of memory patients who have a distribution of memory errors that is Normal, μ = 60? After the treatment they have an average score of = 55, s = 8 memory errors. -1.753 = t crit t obs =-2.5 - Reject H 0 H 0 : μ Treatment > μ pop = 60 H A : μ Treatment < μ pop = 60

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PSY 340 Statistics for the Social Sciences Assumptions of t-statistics Values in the sample must be independent observations –Each observation is independent of all of the other observations (see pg 253 of your textbook for examples of non-independence) The population that is sampled must be normally distributed –It turns out that t-tests are pretty robust against violations of this assumption, especially with large samples

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PSY 340 Statistics for the Social Sciences Effect Sizes & Power for 1-sample t Test Recall for the 1-sample z-test: Remember we don’t know these with the t-test design So for the 1-sample t-test we estimate the Cohen’s d: sample mean Pop mean from null hypothesis sample standard deviation

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PSY 340 Statistics for the Social Sciences Using SPSS: 1 sample t Entering the data –Observations go into one column e.g., 2, 6, 5, 9 Performing the analysis –Analyze -> Compare means -> one sample t-test –Identify which column to do your test on –Enter the ‘Test value’ – this is the population mean in the H 0 –Be careful, the default is 0, this may not be what you want Reading the output –Mean of the sample, the computed-t, degrees of freedom, p-value (Sig.) (also reminds you of your test value)

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PSY 340 Statistics for the Social Sciences Next time Related samples t-tests Independent samples t-tests

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