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

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

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

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

Testing Hypotheses Hypothesis testing: a five step program 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

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 patients 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: HA:

Performing your statistical test What are we doing when we test the hypotheses? Real world (‘truth’) H0: is true (no treatment effect) One population XA the memory treatment sample are the same as those in the population of memory patients. H0: is false (is a treatment effect) Two populations XA they aren’t the same as those in the population of memory patients

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

Performing your statistical test One sample z One sample t identical Test statistic

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

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

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

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-table

One sample t-test The t-statistic distribution (a transformation of the distribution of sample means transformed) To reject the H0, you want a computed test statistics that is large The alpha level gives us the decision criterion New table: the t-table Distribution of the t-statistic If test statistic is here Reject H0 If test statistic is here Fail to reject H0

One sample t-test α - level New table: the t-table One tailed - or - Two-tailed Degrees of freedom df Critical values of t tcrit

One sample t-test What is the tcrit 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 tcrit = +2.571

One sample t-test What is the tcrit for a one-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 n = 6 One-tailed df = n - 1 = 5 tcrit = +2.015

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

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

One sample t-test An example: One sample t-test Memory experiment example: H0: μTreatment > μpop = 60 HA: μTreatment < μpop = 60 We give a n = 16 memory patients a memory improvement treatment. Step 2: Set your decision criteria One -tailed After the treatment they have an average score of = 55, s = 8 memory errors. α = 0.05 How do they compare to the general population of memory patients who have a distribution of memory errors that is Normal, μ = 60?

One sample t-test An example: One sample t-test Memory experiment example: H0: μTreatment > μpop = 60 HA: μTreatment < μpop = 60 We give a n = 16 memory patients a memory improvement treatment. Step 2: Set your decision criteria After the treatment they have an average score of = 55, s = 8 memory errors. One -tailed α = 0.05 How do they compare to the general population of memory patients who have a distribution of memory errors that is Normal, μ = 60?

One sample t-test An example: One sample t-test Memory experiment example: H0: μTreatment > μpop = 60 HA: μTreatment < μpop = 60 We give a n = 16 memory patients a memory improvement treatment. One -tailed α = 0.05 Step 3: Collect your data After the treatment they have an average score of = 55, s = 8 memory errors. How do they compare to the general population of memory patients who have a distribution of memory errors that is Normal, μ = 60?

One sample t-test An example: One sample t-test Memory experiment example: H0: μTreatment > μpop = 60 HA: μTreatment < μpop = 60 We give a n = 16 memory patients a memory improvement treatment. One -tailed α = 0.05 After the treatment they have an average score of = 55, s = 8 memory errors. Step 4: Compute your test statistics How do they compare to the general population of memory patients who have a distribution of memory errors that is Normal, μ = 60? = -2.5

One sample t-test An example: One sample t-test Memory experiment example: H0: μTreatment > μpop = 60 HA: μTreatment < μpop = 60 We give a n = 16 memory patients a memory improvement treatment. One -tailed α = 0.05 t = -2.5 After the treatment they have an average score of = 55, s = 8 memory errors. Step 4: Compute your test statistics How do they compare to the general population of memory patients who have a distribution of memory errors that is Normal, μ = 60?

One sample t-test An example: One sample t-test Memory experiment example: H0: μTreatment > μpop = 60 HA: μTreatment < μpop = 60 We give a n = 16 memory patients a memory improvement treatment. One -tailed α = 0.05 After the treatment they have an average score of = 55, s = 8 memory errors. 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? tcrit = -1.753

One sample t-test An example: One sample t-test Memory experiment example: H0: μTreatment > μpop = 60 HA: μTreatment < μpop = 60 We give a n = 16 memory patients a memory improvement treatment. One -tailed α = 0.05 After the treatment they have an average score of = 55, s = 8 memory errors. 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? tobs=-2.5 - Reject H0 -1.753 = tcrit

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

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: Pop mean from null hypothesis sample mean sample standard deviation

Using SPSS: 1 sample t Entering the data Performing the analysis 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 H0 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)

Next time Related samples t-tests Independent samples t-tests