One Sample t Tests Karl L. Wuensch Department of Psychology East Carolina University

Nondirectional Test Null:  = some value Alternative:   that value We have a sample of N scores Somehow we magically know the value of the population  We trust that the population is normally distributed Or invoke the Central Limit Theorem

H 0 :  IQ = 100 N = 25, M = 107,  = 15 p =.0198, two-tailed

Directional Test For z = 2.33 If predicted direction in H 1 is correct, then p =.0099 If predicted direction in H 1 is not correct, then p = =.9901

Confidence Interval

The Fly in the Ointment How could we know the value of  but not know the value of  ?

Student’s t The sampling distribution of  2 is unbiased but positively skewed. Thus, more often than not, s 2 <  2 And | t | > | z |, giving t fat tails (high kurtosis)

Fat-Tailed t Because of those fat tails, one will need go out further from the mean to get to the rejection region. How much further depends on the df, which are N-1. The fewer the df, the further out the critical values. As df increase, t approaches the normal distribution.

CV for t,  =.05, 2-tailed Degrees of FreedomCritical Value for t  1.960

William Gosset

SAT-Math For the entire nation, between 2000 and 2004,  = 516. For my students in undergrad stats:  M =  s =  N = 114 H 0 : For the population from which my students came,  = 516.

We Reject That Null df = N – 1 = 113 p =.034

CI.95 From the t table for df = 100, CV =

Effect Size Estimate by how much the null is wrong. Point estimate = M – null value Can construct a CI. For our data, take the CI for M and subtract from each side the null value [ – 516, – 516] = [1.43, 36.13]

Standardized Effect Size When the unit of measure is not intrinsically meaningful, As is often case with variables studied by psychologists, Best to estimate the effect size in standard deviation units. The parameter is

Estimated  We should report a CI for  Constructing it by hand in unreasonably difficult. Professor Karl will show how to use SAS or SPSS to get the CI.

Assumptions Only one here, that the population is normally distributed. If that is questionable, one might use nonlinear transformations, especially if the problem is skewness. Or, use analyses that make no normality assumption (nonparametrics and resampling statistics).

Summary Statements who or what the research units were (sometimes called “subjects” or “participants”) what the null hypothesis was (implied) descriptive statistics such as means and standard deviations whether or not you rejected the null hypothesis

Summary Statements 2 if you did reject the null hypothesis, what was the observed direction of the difference between the obtained results and those expected under the null hypothesis what test statistic (such as t) was employed the degrees of freedom

Summary Statements 3 if not obtainable from the degrees of freedom, the sample size the computed value of the test statistic the p value (use SPSS or SAS to get an exact p value) an effect size estimate and a confidence interval for the effect size parameter

Example Summary Statements Carefully study my examples in my document One Mean Inference.One Mean Inference Pay special attention to when and when not to indicate a direction of effect. and also when the CI would more appropriately be with confidence coefficient (1 - 2  ) rather than (1 -  ).

The t Family