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One Sample t-test



Review of Z-test

Used to compare sample mean to aknown population, for which you have

mu and sigma. Enables you to determine whether

difference between sample and

population means is due to chance.



Introducing the t-test

Used when σ is not known and must be estimated

using sample standard deviation (s).

The t-statistic is a substitute for z whenever σ is

unknown New error term:

SM = Estimated standard error

Provides an estimate of the average distance

between a sample mean and the population mean

t-test has its own table of critical values

There are different types of t-tests (one sample,independent samples, and dependent samples)



One Sample t-test Formula



One Sample t-test Example

Research Question: Do birds findstaring aversive?

State Statistical Hypothesis: H0: µplain side = 30 min

H1: µplain side 30 min

Sample Descriptive Statistics: M=35, s=9, n=16

Compute standard error of estimate:

n

s s

X 25.2

16

9

X s



Set decision Criteria:

two-tailed test (nondirectional)

Critical values in t-table based on df = n-1

For our sample, n=16, therefore df=16-1= 15 If α=.05, from t-table, tcrit = + 2.131

Compute t-test statistic:

Make decision: Reject Ho if tobtained > tcrit For our example, reject Ho because tobtained = 2.22 > tcrit = 2.131


X s

x t

22.225.2

3035

t



The t-distribution



t test Critical Values versus ztest Critical Values



t table



Example Repeated two-tailed(Directional Test)

Research Hypothesis: Birds avoideyespots.

State Statistical Hypothesis: H0: µplain side < 30 min

H1: µplain side > 30 min

Sample Descriptive Statistics: M=35, s=9, n=16

Compute standard error of estimate:

n

s s

X 25.2

16

9

X s



Set decision Criteria:

one-tailed test (directional)

Critical values in t-table based on df = n-1

For our sample, n=16, therefore df=16-1= 15 If α=.05, from t-table, tcrit = 1.753

Compute t-test statistic:

Make decision: Reject Ho if tobtained > tcrit

For our example, reject Ho because tobtained = 2.22 > tcrit = 1.753


X s

x

t

22.225.2

3035

t



Writing-up Test Results in APA Format



Writing-up Test Results in APA Format

Birds spent a significantly greater amount of time on the plain side (M =35) compared to

the spotted side (M =25) of the chamber,t(15) = 2.13, p < .05.



What does p<.05 mean?



Assumptions of the t-test

Same as for the z-test:

Independent Observations

Normality

For small samples (e.g., if n < 30),

violations are a problem and affect thevalidity of the hypothesis test. But if sample size is sufficiently large (e.g., if n>30),moderate violations are not a bigproblem.



How big is the effect?

We can use Cohen’s d to estimate effect size:

56.9

3035

s

x

d

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