T tests comparing two means t tests comparing two means.

t testst testscomparing two meanscomparing two means

Overall PurposeOverall Purpose

A t-test is used to compare two A t-test is used to compare two average scores.average scores.

Sample data are used to Sample data are used to answer a question about answer a question about population means.population means.

The population means and The population means and standard deviations are not standard deviations are not known.known.

The Three TypesThe Three Types

There are three ways to use There are three ways to use a a t-test in a comparative t-test in a comparative educational study:educational study:

One sample t-testOne sample t-test Independent t-testIndependent t-test Dependent t-testDependent t-test

Independent t-testIndependent t-test

To compare two groups measured To compare two groups measured at the same time.at the same time.

Treatment vs. Control at pre-testTreatment vs. Control at pre-test Treatment vs. Control at post-testTreatment vs. Control at post-test Males vs. Females - ed. outcomesMales vs. Females - ed. outcomes Hi vs. Lo coping skills – stress Hi vs. Lo coping skills – stress

outcomesoutcomes

One sample t-testOne sample t-test

To compare a sample to a To compare a sample to a population, or one group population, or one group measured at one point in time.measured at one point in time.

Local Sample vs. national normLocal Sample vs. national norm Sample mean vs. some Sample mean vs. some

theoretically or practically theoretically or practically meaningful cut pointmeaningful cut point

Dependent t-testDependent t-test

To compare one group to itself To compare one group to itself over time, or one group measured over time, or one group measured at two times.at two times.

Pre-test vs. post-test for one groupPre-test vs. post-test for one group The mean performance for the The mean performance for the

same group under two different same group under two different drug dosage levelsdrug dosage levels

Assumptions and ConditionsAssumptions and Conditions

NormalityNormality Homogeneity of VarianceHomogeneity of Variance Independence of Independence of

ObservationsObservations Random SamplingRandom Sampling Population Variance Not Population Variance Not

KnownKnown

Testing AssumptionsTesting Assumptions

NormalityNormality– Histograms and boxplotsHistograms and boxplots– Visual inspectionVisual inspection– Reasonably within sampling error of normalReasonably within sampling error of normal

Homogeneity of VarianceHomogeneity of Variance– SPSS gives you a testSPSS gives you a test– p < .05 means assumption not metp < .05 means assumption not met– If not met, you separate variance estimate If not met, you separate variance estimate

t-test t-test

Identifying ConditionsIdentifying Conditions

Independence of ObservationsIndependence of Observations– No score influences another scoreNo score influences another score– Study design featureStudy design feature

Random SamplingRandom Sampling– This is a formal assumptionThis is a formal assumption– It is also a study design featureIt is also a study design feature

Population Variance Not KnownPopulation Variance Not Known– Sample data is used to make estimates Sample data is used to make estimates

of population parametersof population parameters– Exception is the test value for the One Exception is the test value for the One

Sample t-testSample t-test

Additional ConsiderationsAdditional Considerations

Confidence intervals can Confidence intervals can be used to test the same be used to test the same hypotheses.hypotheses.

There is a unique critical There is a unique critical t value for each degrees t value for each degrees of freedom condition.of freedom condition.


Random assignment is needed Random assignment is needed to make causal inferences when to make causal inferences when using the independent t-test.using the independent t-test.

If intact groups are compared, If intact groups are compared, examine differences on examine differences on potential confounding variables.potential confounding variables.

Statistical SignificanceStatistical Significance

How do you know when How do you know when there is a statistically there is a statistically significant difference significant difference between the average between the average scores you are scores you are comparing?comparing?


When the p value is less When the p value is less than alpha, usually set than alpha, usually set at .05.at .05.

What does a small p What does a small p value mean?value mean?


You are most interested You are most interested in whether the amount of in whether the amount of difference you observe in difference you observe in your sample means is your sample means is more than would be more than would be expected due to sampling expected due to sampling error alone.error alone.


The z value of 1.96 serves as a The z value of 1.96 serves as a rough guideline for evaluating a rough guideline for evaluating a

t value.t value. It means that the amount of It means that the amount of

difference in the means is difference in the means is approximately twice as large as approximately twice as large as expected due to sampling error expected due to sampling error alone.alone.

Interpret the p value.Interpret the p value.


If the two population means are If the two population means are equal, your sample data can still equal, your sample data can still show a difference due to sampling show a difference due to sampling error.error.

The p value indicates the probability The p value indicates the probability of results such as those obtained, or of results such as those obtained, or larger, given that the null hypothesis larger, given that the null hypothesis is true and only sampling error has is true and only sampling error has lead to the observed difference.lead to the observed difference.


You have to decide which is a You have to decide which is a more reasonable conclusion:more reasonable conclusion:

There is a real difference There is a real difference between the population means.between the population means.OrOr

The observed difference is due The observed difference is due to sampling error.to sampling error.


We call these We call these conclusions:conclusions:

Rejecting the null Rejecting the null hypothesis.hypothesis.

Failing to reject the null Failing to reject the null hypothesis.hypothesis.


If the p value is small, less If the p value is small, less than alpha (typically set than alpha (typically set at .05), then we conclude at .05), then we conclude that the observed sample that the observed sample difference is unlikely to be difference is unlikely to be the result of sampling error.the result of sampling error.


If the p value is large, If the p value is large, greater than or equal to greater than or equal to alpha (typically set at .05), alpha (typically set at .05), then we conclude that the then we conclude that the difference you observed difference you observed could have occurred by could have occurred by sampling error, even when sampling error, even when the null hypothesis is true.the null hypothesis is true.

HypothesesHypotheses

Hypotheses for the Independent t-testHypotheses for the Independent t-testNull Hypothesis:Null Hypothesis: 11 = = 2 2 or or 11 - - 2 2 = 0= 0Directional Alternative Hypothesis:Directional Alternative Hypothesis: 11 > > 22 or or 11 - - 2 2 > 0> 0Non-directional Alternative Hypothesis:Non-directional Alternative Hypothesis: 11 =/= =/= 2 2 or or 11 - - 2 2 =/= 0=/= 0where:where: 11 = population mean for group one = population mean for group one 2 2 = population mean for group two= population mean for group two

HypothesesHypotheses Hypotheses for the One Sample t-testHypotheses for the One Sample t-testNull Hypothesis:Null Hypothesis: = = 0 0 or or - - 0 0 = 0= 0Directional Alternative Hypothesis:Directional Alternative Hypothesis: > > 00 or or - - 0 0 > 0> 0Non-directional Alternative Hypothesis:Non-directional Alternative Hypothesis: =/= =/= 0 0 or or - - 0 0 =/= 0=/= 0where:where: = population mean for group of interest (local = population mean for group of interest (local

population)population) 0 0 = population mean for comparison (national = population mean for comparison (national

norm)norm)

ExampleExample

Our sample research design:Our sample research design:Head Start children randomly assigned Head Start children randomly assigned

to get in the program or be placed on to get in the program or be placed on the program’s waiting list.the program’s waiting list.

Pre and Post test data collected on Pre and Post test data collected on cognitive, social, and physical cognitive, social, and physical developmental outcomesdevelopmental outcomes

Measures scales with mean=50, SD=10.Measures scales with mean=50, SD=10.

ExampleExample

Our sample research design:Our sample research design:The subjects were not randomly sampled. The subjects were not randomly sampled.

They are volunteers. The parents of the They are volunteers. The parents of the study children applied for admission to study children applied for admission to Head Start in neighborhoods where more Head Start in neighborhoods where more children apply than can be served.children apply than can be served.

There are no other publicly funded There are no other publicly funded preschool programs in the area for low preschool programs in the area for low income children.income children.

ExampleExample

Our sample research design:Our sample research design:

Pre PostPre-schoolAt home

T tests comparing two means t tests comparing two means.

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