t-Testt-Test

Comparing Means From Two Sets Comparing Means From Two Sets of Dataof Data

Steps For Comparing GroupsSteps For Comparing Groups

Assumptions of t-TestAssumptions of t-Test

Dependent variables are interval or ratio.Dependent variables are interval or ratio.The population from which samples are The population from which samples are drawn is normally distributed.drawn is normally distributed.Samples are randomly selected.Samples are randomly selected.The groups have equal variance The groups have equal variance (Homogeneity of variance).(Homogeneity of variance).The t-statistic is robust (it is reasonably The t-statistic is robust (it is reasonably reliable even if assumptions are not fully reliable even if assumptions are not fully met.met.

Computing Confidence IntervalsComputing Confidence Intervals

We can determine the probability that a population mean We can determine the probability that a population mean lies between certain limits using a sample mean.lies between certain limits using a sample mean.With inferential statistics we reverse this process and With inferential statistics we reverse this process and determine the probability that a random sample drawn determine the probability that a random sample drawn from a specific population would differ by an observed from a specific population would differ by an observed result.result.

t Valuest Values

Critical value Critical value decreases if N is decreases if N is increased.increased.Critical value Critical value decreases if alpha decreases if alpha is increased.is increased.Differences Differences between the means between the means will not have to be will not have to be as large to find sig as large to find sig if N is large or if N is large or alpha is increased.alpha is increased.

Probability that a sample came from a population?Probability that a sample came from a population?

Using the standard error we compute the probability that Using the standard error we compute the probability that two means come from the same population.two means come from the same population.If Z or t exceed the level of significance we conclude that If Z or t exceed the level of significance we conclude that the sample wasthe sample was

Not drawn from the population orNot drawn from the population or Has been modified so that it no longer represents the populationHas been modified so that it no longer represents the population

Relationship between t Statistic and PowerRelationship between t Statistic and Power

To increase power:To increase power: Increase the difference Increase the difference

between the means.between the means. Reduce the varianceReduce the variance Increase NIncrease N Increase α from α = .01 to Increase α from α = .01 to

α = .05α = .05

Does Volleyball Serve Training Improve Serving Ability?Does Volleyball Serve Training Improve Serving Ability?

Population mean = 31, sd = Population mean = 31, sd = 7.5.7.5.30 students given serve 30 students given serve training. Following training training. Following training mean = 35, sd = 8.3.mean = 35, sd = 8.3.Critical Z = 1.96Critical Z = 1.96Probability is greater than Probability is greater than 99 to 1 that the mean did 99 to 1 that the mean did not come from original not come from original population.population.The training was effective.The training was effective.

Volleyball Example Using t-statisticVolleyball Example Using t-statistic

Critical value of t(29)= 2.045, p = 0.05Critical value of t(29)= 2.045, p = 0.05

Since obtained t > critical value these Since obtained t > critical value these means are statistical different.means are statistical different.

Comparing Two Independent SamplesComparing Two Independent Samples

Independent samples (males, females), Independent samples (males, females), (swimmers, runners).(swimmers, runners).

Must be different subjects in each group.Must be different subjects in each group.

Independent t TestIndependent t Test

If the t statistic is greater than the critical If the t statistic is greater than the critical value wevalue we

Conclude the independent variable had a Conclude the independent variable had a significant effectsignificant effect

And we reject chance as the cause of the And we reject chance as the cause of the mean difference.mean difference.

Effects of Verbal Lesson of Basketball Shooting SkillEffects of Verbal Lesson of Basketball Shooting Skill

Critical value of t(120) = 1.98, p = 0.05

Since our obtained t(98) = -1.36 is NOT greater than the critical value we ACCEPT the Null Hypothesis. The training had no effect upon shooting skill.

Note: The sign +/- of t does not matter.

Does Positive Does Positive Reinforcement Reinforcement Affect Bowling?Affect Bowling?

Critical value t(40) = 2.201, p = 0.05

Since obtained t > critical t

We reject the Null and state that positive reinforcement significantly

improves bowling ability.

Summary Table for Effects of Praise on BowlingSummary Table for Effects of Praise on Bowling

The t-test With Unequal NThe t-test With Unequal N

When you have unequal numbers of subjects in each group the statistic uses a different equation to

estimate the standard error of the differences between groups.

The t-test With Unequal NThe t-test With Unequal N

Critical value of t(16) = 2.120, p = .05. The groups are significantly different.

Dependent or Paired t-testDependent or Paired t-test

The Dependent t-test is more powerful that the Independent Groups t-test.

Note that the equation uses the correlation between pre and post samples.

Dependent or Paired t-testDependent or Paired t-test The same subjects are in each group

(DEPENDENT or PAIRED t-test).

Critical value t(29) = 2.045, p = 0.05

The groups ARE SIGNIFICANTLY

Different.

Note: the correction formula adjusts the

variance between groups. Since the same subjects are in each group you can

expect less variance.

Repeated Measures experiments are more

powerful than independent groups

Does a Bicycle Tour Affect Self-Esteem?Does a Bicycle Tour Affect Self-Esteem?Are these differences MEANINGFUL????Are these differences MEANINGFUL????

Critical value of t(60) = 2.000, p = 0.05, so there is a significant difference. BUT DOES IT MEAN ANYTHING???

The Magnitude of the Difference (Size of Effect)The Magnitude of the Difference (Size of Effect)

Omega squared can be used to determine the Omega squared can be used to determine the importance, or usefulness of the meanimportance, or usefulness of the mean difference. difference.

ωω2 2 is the percentage of the variance (diff between is the percentage of the variance (diff between means) that can be explained by the independent means) that can be explained by the independent variable.variable.

In this case the low-back and hip study explains In this case the low-back and hip study explains 21% of variance between the means (pre & post).21% of variance between the means (pre & post).

Cohen’s Effect SizeCohen’s Effect Size

Effect size of .2 is small, Effect size of .2 is small, .5 moderate.5 moderate, , .8 large.8 large

The control group is used to compute SD The control group is used to compute SD because it is not contaminated by the treatment because it is not contaminated by the treatment effect.effect.

The Percent Change is also useful in evaluating if a change The Percent Change is also useful in evaluating if a change is meaningful.is meaningful.

Before doing an experiment you should know Before doing an experiment you should know what what Percent ChangePercent Change would be considered would be considered meaningfulmeaningful..

For an Olympic athlete, a 1% (For an Olympic athlete, a 1% (meaningfulmeaningful) ) improvement can be the difference between improvement can be the difference between winning and losing.winning and losing.

For an untrained individual a 1% improvement For an untrained individual a 1% improvement would probably be would probably be meaninglessmeaningless..

PracticalPractical & & MeaningfulMeaningful Significance Significance

If two means are significantly different, that does If two means are significantly different, that does not imply that they are practical.not imply that they are practical.

If two means are NOT statistically significant, If two means are NOT statistically significant, that does not imply that their differences are not that does not imply that their differences are not practical.practical.

Use Use ωω22, , Effect SizeEffect Size and and Percent ChangePercent Change to to evaluate the meaningfulness of an outcome.evaluate the meaningfulness of an outcome.

Type I and Type II ErrorsType I and Type II Errors

Type I Error: Stating that there is a difference when there isn’t.

Type II Error: Stating there is no difference when there is one.

We can never know if we have made a Type I or II error.We can never know if we have made a Type I or II error.

Statistics only provide the probability of making a Type I or Statistics only provide the probability of making a Type I or II error.II error.

The critical factor in this decision is the consequence of The critical factor in this decision is the consequence of being wrong.being wrong.

The confidence level should be set to protect against the The confidence level should be set to protect against the most costly error.most costly error.

Which is worse: to accept the null hypothesis when it is Which is worse: to accept the null hypothesis when it is really false or to reject it when it is really true?really false or to reject it when it is really true?

Two Tailed Test: Null No Difference.Two Tailed Test: Null No Difference.

One Tail Test: Null A > B. More Powerful, easier to find One Tail Test: Null A > B. More Powerful, easier to find differences.differences.

Power: the ability to detect differences if they exist.Power: the ability to detect differences if they exist.

Statistical PowerStatistical Power

Power ( 1 - β ) depends upon:Power ( 1 - β ) depends upon:

1.1. Alpha [ZAlpha [Zαα (.10) = 1.65, Z (.10) = 1.65, Zαα (.05) = 1.96] (.05) = 1.96]

2.2. Difference between the means.Difference between the means.

3.3. Standard deviations between the two Standard deviations between the two groups.groups.

4.4. Sample size N. Sample size N.

To Increase PowerTo Increase Power

Increase alpha, Power for α = .10 is Increase alpha, Power for α = .10 is greater than power for α = .05greater than power for α = .05

Increase the difference between means.Increase the difference between means.

Decrease the sd’s of the groups.Decrease the sd’s of the groups.

Increase N.Increase N.

Calculation of PowerCalculation of Power

In this example

Power (1 - β ) = 70.5%

From Table A.1 Zβ of .54 is 20.5%

Power is

20.5% + 50% = 70.5%

Calculation of Sample Calculation of Sample Size to Produce a Size to Produce a

Given PowerGiven Power

Compute Sample Size N for a Power of .80 at p = 0.05

The area of Zβ must be 30% (50% + 30% = 80%) From Table A.1 Zβ = .84

If the Mean Difference is 5 and SD is 6 then 22.6 subjects would be required to have a power of .80

Calculation of Sample Sized Need to Obtain a Desired Calculation of Sample Sized Need to Obtain a Desired Level of PowerLevel of Power

PSD 30 Newtons

Alpha 1.96 this is p=.05

Beta

80 0.84 these are beta values

90 1.28

95 1.645

Power

Stdev 80 90 95

30 16 21 26These values in red are the N

needed based on your PSD.

20 7 9 12

10 2 2 3

The boxed values are values you must input, based on previous literature.

PSD = Practical Significant Difference

PowerPower

Research performed with insufficient Research performed with insufficient power may result in a Type II error,power may result in a Type II error,Or waste time and money on a study that Or waste time and money on a study that has little chance of rejecting the null.has little chance of rejecting the null.In power calculation, the values for mean In power calculation, the values for mean and sd are usually not known beforehand.and sd are usually not known beforehand.Either do a PILOT study or use prior Either do a PILOT study or use prior research on similar subjects to estimate research on similar subjects to estimate the mean and sd.the mean and sd.

Independent t-TestIndependent t-Test

For an Independent t-Test you need a

grouping variable to define the groups.

In this case the variable Group is

defined as

1 = Active

2 = Passive

Use value labels in SPSS

Independent t-Test: Defining Independent t-Test: Defining VariablesVariables

Grouping variable GROUP, the level of measurement is Nominal.

Be sure to enter value

labels.

Independent t-TestIndependent t-Test

Independent t-Test: Independent & Independent t-Test: Independent & Dependent VariablesDependent Variables

Independent t-Test: Define GroupsIndependent t-Test: Define Groups

Independent t-Test: OptionsIndependent t-Test: Options

Independent t-Test: OutputIndependent t-Test: OutputGroup Statistics

10 2.2820 1.24438 .39351

10 1.9660 1.50606 .47626

GroupActive

Passive

Ab_ErrorN Mean Std. Deviation

Std. ErrorMean

Independent Samples Test

.513 .483 .511 18 .615 .31600 .61780 -.98194 1.61394

.511 17.382 .615 .31600 .61780 -.98526 1.61726

Equal variancesassumed

Equal variancesnot assumed

Ab_ErrorF Sig.

Levene's Test forEquality of Variances

t df Sig. (2-tailed)Mean

DifferenceStd. ErrorDifference Lower Upper

95% ConfidenceInterval of the

Difference

t-test for Equality of Means

Assumptions: Groups have equal variance [F = .513, p =.483, YOU DO NOT WANT THIS TO

BE SIGNIFICANT. The groups have equal variance, you have not violated an assumption

of t-statistic.

Are the groups different?

t(18) = .511, p = .615

NO DIFFERENCE

2.28 is not different from 1.96

Dependent or Paired t-Test: Define Dependent or Paired t-Test: Define VariablesVariables

Dependent or Paired t-Test: Select Paired-Dependent or Paired t-Test: Select Paired-SamplesSamples

Dependent or Paired t-Test: Select Dependent or Paired t-Test: Select VariablesVariables

Dependent or Paired t-Test: OptionsDependent or Paired t-Test: Options

Dependent or Paired t-Dependent or Paired t-Test: OutputTest: Output

Paired Samples Statistics

4.7000 10 2.11082 .66750

6.2000 10 2.85968 .90431

Pre

Post

Pair1

Mean N Std. DeviationStd. Error

Mean

Paired Samples Correlations

10 .968 .000Pre & PostPair 1N Correlation Sig.

Paired Samples Test

-1.50000 .97183 .30732 -2.19520 -.80480 -4.881 9 .001Pre - PostPair 1Mean Std. Deviation

Std. ErrorMean Lower Upper

95% ConfidenceInterval of the

Difference

Paired Differences

t df Sig. (2-tailed)

Is there a difference between pre & post?

t(9) = -4.881, p = .001

Yes, 4.7 is significantly different from 6.2