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Statistics for the Social Sciences Psychology 340 Fall 2006 Using t-tests

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Statistics for the Social Sciences Outline (for next 3 lectures) Review t-tests –One sample, related samples, independent samples –Additional assumptions Levene’s test

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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 ( ) and standard deviation ( )are known

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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 ( ) is known –but standard deviation ( ) is NOT known

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

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

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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 -level of 0.05? Distribution of the t-statistic = 0.05 Two-tailed n = 6 df = n - 1 = 5 t crit = + 2.571

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Statistics for the Social Sciences One sample t-test Distribution of the t-statistic = 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 -level of 0.05?

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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, = 60? After the treatment they have an average score of = 55, s = 8 memory errors. Don’t know Do know s

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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, = 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 Treatment > pop = 60 Treatment < pop = 60

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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 : Treatment > pop = 60 H A : Treatment < pop = 60 = 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, = 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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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 H 0 : Treatment > pop = 60 H A : Treatment < pop = 60 = 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, = 60? After the treatment they have an average score of = 55, s = 8 memory errors.

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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. = 0.05 One -tailed Step 3: Collect your data H 0 : Treatment > pop = 60 H A : Treatment < pop = 60 How do they compare to the general population of memory patients who have a distribution of memory errors that is Normal, = 60? After the treatment they have an average score of = 55, s = 8 memory errors.

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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. = 0.05 One -tailed Step 4: Compute your test statistics = -2.5 H 0 : Treatment > pop = 60 H A : Treatment < pop = 60 How do they compare to the general population of memory patients who have a distribution of memory errors that is Normal, = 60? After the treatment they have an average score of = 55, s = 8 memory errors.

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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. = 0.05 One -tailed Step 4: Compute your test statistics t = -2.5 H 0 : Treatment > pop = 60 H A : Treatment < pop = 60 How do they compare to the general population of memory patients who have a distribution of memory errors that is Normal, = 60? After the treatment they have an average score of = 55, s = 8 memory errors.

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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. = 0.05 One -tailed Step 5: Make a decision about your null hypothesis H 0 : Treatment > pop = 60 H A : Treatment < pop = 60 How do they compare to the general population of memory patients who have a distribution of memory errors that is Normal, = 60? After the treatment they have an average score of = 55, s = 8 memory errors. t crit = -1.753

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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. = 0.05 One -tailed Step 5: Make a decision about your null hypothesis H 0 : Treatment > pop = 60 H A : Treatment < pop = 60 How do they compare to the general population of memory patients who have a distribution of memory errors that is Normal, = 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

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Statistics for the Social Sciences Next time Related samples t-tests Independent samples t-tests Using SPSS with t-tests (maybe next Monday)

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