Week 3

t-Tests

  • Assess whether two means are statisically different

  • 3 types:

    • One-sample t-tests compare a sample mean to a single value

      • e.g., comparing a sample to a known (or hypothesized) population mean

    • Independent-sample t-tests compare the means of two independent groups

      • Between-subjects designs

    • Paired-sample t-tests compare the means of two non-independent groups

      • Within-subjects designs

  • Depend on the Central Limit Theorem (CLT)

Central Limit Theorem (CLT)

  • If you repeatedly, randomly sample from a popualtion with mean μ, then the distribution of the sample means will be centered on μ and approximately normally distributed

  • The standard deviation of the sampling distribution will depend on N

t Distributions

  • The CLT lets us characterize a series of t Distributions with known characteristics (mean = μ; standard deviation that depends on N)

  • With a known N and a known x̄, we can calculate how far away Xis from the center of the relevant t distribution

  • This yields a t value

  • Under the assumption that our sample(s) really do come from a distribution centered on μ, more extreme t values are less likely

  • Functionally this is a test of H0!

  • When there’s less than a 5% chance of obtaining our observed t value, we conclude that our sample(s) probably aren’t taken from a distribution cnetered on μ - that is the means differ

  • Note: with α = .05, our cut-off is a 2.5% likelihood in each tail (5% in total – this is a “two-tailed” test)

  • If our t value is more extreme than the “critical t values” at these cut-off points, then we reject H0

  • We’ll also get a p value that tells us exactly how likely our t value was under H0

One-Sample t-Tests

  • Compare a sample mean to a single value

    • Does the mean IQ of UIowa undergraduates differ from 100?

    • Do pigeons’ test scores differ from chance (50%)?

  • t-Tests are parametric, meaning that they rest on certain assumptions

    • (necessary to construct the underlying t distribution)

  • We have to check these assumptions before we run the t-Test

Steps for a One-Sample t-Test

  • Check the assumptions

    • The DV must be continuous

    • The observation must be independent

      • This is more about study design; not something we’ll test

    • The DV must be (approximately) normal

      • We’ll test for significant depatures of normality using a “Shapiro-Wilk” test

  • Calculate the t and p values

  • Report in standardized format

    • The sample mean[did / did not] significantly differ from [hypothesized value], t(df) = [], p = []

    • Where df (degrees of freedom) = N - 1

Assumptions of One-Sample t-Tests

  • If the assumptions are met, then we can run the t-Test

  • If the assumptions aren’t met, then we can’t run the t-test

    • Move to “non-parametric” stats, which are (a bit) more complicated but require fewer assumptions

    • We won’t deal with non-paraetric stats in this course