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 5th 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

SymmetricUnimodalAsymptotic

Slightly different shape than the normal curve, meaning that the critical values are similar but slightly larger.

Comparing critical values

One-tail Two-tail df = One-tail Two-tail

p < .05 10 p < .05

p < .01 p < .01

25 p < .05

p < .01

100 p < .05

p < .01

p < .05

p < .01

z distribution t distribution

Four Steps

1. State the hypothesisH0 vs. HAlternate

2. Identify your criterion for rejecting H0

  Directional or non-directional testOne-tailed or two-tailed

 

Set alpha level (Prob. incorrectly rejecting H0)

3. Compute test statistic General form: Test = statistic – expected parameter

standard error of statistic

4. Decide about H0

Example: One Sample

• Random sample of 9 subjects from freshman population. Mean score was 625 with SD of 90. Ho: =500, H1: 500 =.05

17.4

990

500625

xs

xt

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 = 2.306

=.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.

xdfcritical stx *),2/(

Two Independent Means

• Review:

2121

)21()21(

xxxx

xxz

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

0men women

Testing:

Assumptions: Normality, Equality of variance, sample sizes equal

men=20, women=25, smen=5, swomen=5, nmen=20, nwomen=20

205

205

0)2520(

22t = df=20+20-2=38

Example: Two-Sample

0men women

Testing:

Assumptions: Normality, Equality of variance, NOTE: sample sizes unequal

men=20, women=25, smen=5, swomen=5, nmen=20, nwomen=30

2

1

1

1221 nn

ss pooledyy

221

)12()11( 22

212

nn

snsns pooled

Example: Two-Sample

0men women

Testing:

men=20, women=25, smen=5, swomen=5, nmen=20, nwomen=30

2

1

1

1221 nn

ss pooledyy

221

)12()11( 22

212

nn

snsns pooled

23020)5(29)5(19

301

201

0)2520(22

t = ? 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?

d

d

s

dt

20,4,6 nsd d

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. 551-572

• One Way ANOVA– Chapter 13 p. 370-384