Education 793 Class Notes T-tests 29 October 2003

Today’s Agenda Class and lab announcements Questions? Hypothesis testing where  needs to be estimated and for dependent samples

Review: One sample tests Uses the general form for testing whether or not the statistic observed is unusual: In practice, can be tested in two ways:  known  estimated

Differences in the Two Tests For the z-test, the critical value comes from a known distribution (normal) and can be obtained directly from a table. In the t-test, the mean and standard error vary from sample to sample. Simulated sample of the t-distribution show that it is like the normal distribution –symmetric –can be interpreted as a probability distribution Unlike the normal distribution, there is a different t-distribution for every sample size greater than 2. The tails of the t-distribution are higher, thus there is more area under the tails than the normal distribution. –Implication is that the critical values for the t-test are slightly higher (the adjustment for sampling variation) than the normal distribution As sample size increases (greater than 30), the t-distribution approximates the normal distribution.

Degrees of Freedom Members of the t-distribution differ based on their degrees of freedom, not sample size. DF=the number of values in the final calculation of a statistic that are free to vary or the number of independent pieces of information. The definition of df varies according to the statistic being calculated. In the case of a mean, df = number of cases – 1. If we have a mean calculated from 5 cases, once we know the values associated with 4 of the cases we can algebraically figure out the value of the 5 th case.

Assumptions of t-test Scores are randomly sampled from some population Scores are normally distributed –How do we examine this. For larger samples, with histograms. For samples greater than 30 we know the t-distribution approaches normality. For smaller samples we use prior information. If prior information is not available, then we may violate the normality assumption in which case non-parametric test are a better option (See later chapters).

Review: z and t z – The normal curve Symmetric Unimodal Asymptotic t – A nearly normal curve Symmetric Unimodal Asymptotic Slightly different shape than the normal curve, meaning that the critical values are similar but slightly larger.

Comparing critical values One-tailTwo-tail df = One-tailTwo-tail p <.0510 p <.05 p < p <.05 p < p <.05 p <.01  p <.05 p <.01 z distribution t distribution

Four Steps 1.State the hypothesis H 0 vs. H Alternate 2.Identify your criterion for rejecting H 0 Directional ornon-directional test One-tailedortwo-tailed Set alpha level (Prob. incorrectly rejecting H 0 ) 3.Compute test statistic General form: Test = statistic – expected parameter standard error of statistic 4.Decide about H 0

Example: One Sample Random sample of 9 subjects from freshman population. Mean score was 625 with SD of 90. Ho:  =500, H1:  500  =.05 What are the degrees of freedom? What is the t-critical value?

Example Reject the Null in favor of the alternative,  not equal to 500  =500 t-critical, df=8 =  =.05 t-observed=4.17

Confidence Intervals Computed the same as with normal curve but replace the critical value with the appropriate critical t-value.

Two Independent Means Review: In this case  ’s are known In order to test if the difference between two independent means is statistically significant we need an estimate of the standard error for the difference. When the variances can be assumed equal:

Assumptions The scores in two groups are randomly sampled from their respective populations The scores in the respective populations are normally distributed The variances are equal (homogeneity of variance)

Example: Two-Sample Testing: Assumptions: Normality, Equality of variance, sample sizes equal  men =20,  women =25, s men =5, s women =5, n men =20, n women =20 t = df= =38

Example: Two-Sample Testing: Assumptions: Normality, Equality of variance, NOTE: sample sizes unequal  men =20,  women =25, s men =5, s women =5, n men =20, n women =30

Example: Two-Sample Testing:  men =20,  women =25, s men =5, s women =5, n men =20, n women =30 = ? df=? Reject ?

Two-Sample: Dependent Means Used to examine data when two observations are made on the same subject or when one observation is made on each of two members of a matched pair. The observations are correlated we therefore do not have two pieces of independent information In our case we will compute the differences for each matched pair and treat them as one sample of differences (Second set of formulas in the chapter)

Example: Dependent Means Ho: No difference in the matched pairs Note we have 20 matched pairs (40 subjects in total) Compute test statistic df=? Reject?

LUCKY US We will have SPSS compute all of these test statistics for us. Laptop exersize if time

Next Week Chi-Square Tests –Chapter 19 p One Way ANOVA –Chapter 13 p