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July+27th+and+28th

July+27th+and+28th - Quantitative Methods in Psychology...

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Quantitative Methods in Psychology July 27th, 2009: t-test week Jeff Vietri Corresponding to chapter 9 in the text

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This week We will cover the three t-tests All of them are based on the same logic as the z-score: How likely are we, given the precision of our measurement, to observe this measurement if there were no relationship between our treatment (IV) and the outcome we're interested in (DV)?
The t-tests Single-sample t-test Used to test whether our sample mean is different from what we would expected from an untreated population. We provide the population mean Independent-samples t-test Used to test whether two sample means, each one drawn from different samples, are different from each other. Paired-samples t-test Used to test whether there is a difference between two measurements taken from the same sample

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Today's Agenda Answer questions about exam/go over problems Introduce t-test Approximation of z-test More useful than z Do single-sample t-tests Introduce logic of independent-samples t- test
The z-test The z-test allowed us to determine how likely we would be to observe a particular sample mean if H 0 were true Calculated z-score using the sample mean, estimated value of the mean, and σ M Looked up corresponding p in unit normal table Made our statistical decision However, we needed to know σ , which we often do not.

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If you don't know a parameter. .. If we don't know a parameter, then what do we do? Estimate it with the sample statistic! This is what we'll do for the t-test Essentially, a t-test is a z-test calculated using standard error estimated with the sample instead of the real parameter
The t-statistic (chapter 9) We’ve been using μ and σ to find out about sampling distributions But about when we don’t know σ? It’s rare to know σ in the real world We can estimate σ with s Use s to estimate sampling distributions M M z μ σ - =

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t-statistic Recall that s and s² are unbiased estimates of σ and σ ² when n-1 is used Thus, s or s ² can be used to estimate the population standard deviation/variance t formula: just like z formula, except it uses s Conceptually, these formulas are the same M M t s μ - = 2 ( ) 1 X M s n - = - 2 2 ( ) 1 X X n s n - = - M M z σ - = 1 SS s n = -
The standard error The standard error of a t-test ( ) is also called: the estimated standard error the standard error of the means Like with z, the standard error is the standard deviation of the distribution of sample means M s 2 M s s s n n = = 2 M t s n μ - = M M t s - =

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The t and z-distributions are similar Both compare Ms to μ in standard deviation units Are sampling distributions Are bell-curve shaped Have a mean of 0
However, the t-distribution has “fatter tails” than the z-distribution, except when n is large In the z-distribution σ is always the same

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July+27th+and+28th - Quantitative Methods in Psychology...

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