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

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

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

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

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

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:

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.

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

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

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

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”

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

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

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)

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

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 =

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 = What is the t crit for a one-tailed hypothesis test with a sample size of n = 6 and an  -level of 0.05?

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

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

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

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.

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.

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.

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.

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 =

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 t obs = Reject H 0

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