Sampling Distributions, Hypothesis Testing and

One-sample Tests

    

Media ViolenceMedia Violence

• Does violent content in a video affect later Does violent content in a video affect later behavior?behavior? Bushman (1998)Bushman (1998)

• Two groups of 100 subjects saw a videoTwo groups of 100 subjects saw a video Violent video versus nonviolent videoViolent video versus nonviolent video

• Then free associated to 26 homonyms with Then free associated to 26 homonyms with aggressive & nonaggressive forms.aggressive & nonaggressive forms. e.g. cuff, mug, plaster, pound, socke.g. cuff, mug, plaster, pound, sock

Cont.

Media Violence--cont.Media Violence--cont.

• ResultsResults Mean number of aggressive free Mean number of aggressive free

associates = 7.10associates = 7.10

• Assume we know that without Assume we know that without aggressive video the mean would aggressive video the mean would be 5.65, and the standard deviation be 5.65, and the standard deviation = 4.5= 4.5 These are parameters (These are parameters (andand

• Is 7.10 enough larger than 5.65 to Is 7.10 enough larger than 5.65 to conclude that video affected conclude that video affected results?results?

Sampling Distribution of Sampling Distribution of the Meanthe Mean

• We need to know what kinds of sample We need to know what kinds of sample means to expect if video has no effect.means to expect if video has no effect. i. e. What kinds of means if i. e. What kinds of means if = 5.65 and = 5.65 and

= 4.5? = 4.5?

This is the sampling distribution of the This is the sampling distribution of the mean.mean.

Cont.

Mean Number Aggressive Associates

7.257.00

6.756.50

6.256.00

5.755.50

5.255.00

4.754.50

4.254.00

3.75

Sampling Distribution

Number of Aggressive AssociatesFr

equ

ency

1400

1200

1000

800

600

400

200

0

Std. Dev = .45

Mean = 5.65

N = 10000.00

Cont.

Sampling Distribution of Sampling Distribution of the Mean--cont.the Mean--cont.

• The sampling distribution of the mean The sampling distribution of the mean depends ondepends on Mean of sampled populationMean of sampled population

• Why?Why?

St. dev. of sampled populationSt. dev. of sampled population• Why?Why?

Size of sampleSize of sample• Why?Why?

Cont.

Sampling Distribution of Sampling Distribution of the mean--cont.the mean--cont.

• Shape of the sampling distributionShape of the sampling distribution Approaches normalApproaches normal

• Why?Why?

Rate of approach depends on sample sizeRate of approach depends on sample size• Why?Why?

• Basic theoremBasic theorem Central limit theoremCentral limit theorem

Central Limit TheoremCentral Limit Theorem

• Given a population with mean = Given a population with mean = and standard deviation = and standard deviation = , the , the sampling distribution of the mean sampling distribution of the mean (the distribution of sample means) (the distribution of sample means) has a mean = has a mean = , and a standard , and a standard deviation = deviation = / /nn. The distribution . The distribution approaches normal as approaches normal as nn, the sample , the sample size, increases.size, increases.

DemonstrationDemonstration• Let population be very skewedLet population be very skewed

• Draw samples of 3 and calculate meansDraw samples of 3 and calculate means

• Draw samples of 10 and calculate meansDraw samples of 10 and calculate means

• Plot meansPlot means

• Note changes in means, standard Note changes in means, standard deviations, and shapesdeviations, and shapes

Cont.

X

20.018.0

16.014.0

12.010.0

8.06.0

4.02.0

0.0

Skewed Population F

req

ue

ncy

3000

2000

1000

0

Std. Dev = 2.43

Mean = 3.0

N = 10000.00

Parent PopulationParent Population

Cont.

Sampling Distribution Sampling Distribution nn = = 33

Sample Mean

13.0012.00

11.0010.00

9.008.00

7.006.00

5.004.00

3.002.00

1.000.00

Sampling Distribution

Sample size = n = 3F

req

ue

ncy

2000

1000

0

Std. Dev = 1.40

Mean = 2.99

N = 10000.00

Cont.

Sampling Distribution Sampling Distribution nn = = 1010

Sample Mean

6.506.00

5.505.00

4.504.00

3.503.00

2.502.00

1.501.00

Sampling Distribution

Sample size = n = 10F

req

ue

ncy

1600

1400

1200

1000

800

600

400

200

0

Std. Dev = .77

Mean = 2.99

N = 10000.00

Cont.

Demonstration--cont.Demonstration--cont.

• Means have stayed at 3.00 throughout--Means have stayed at 3.00 throughout--except for minor sampling errorexcept for minor sampling error

• Standard deviations have decreased Standard deviations have decreased appropriatelyappropriately

• Shapes have become more normal--see Shapes have become more normal--see superimposed normal distribution for superimposed normal distribution for referencereference

Steps in Hypothesis Steps in Hypothesis TestingTesting

• Define the null hypothesis.Define the null hypothesis.

• Decide what you would expect to Decide what you would expect to find if the null hypothesis were true.find if the null hypothesis were true.

• Look at what you actually found.Look at what you actually found.

• Reject the null if what you found is Reject the null if what you found is not what you expected.not what you expected.

The Null HypothesisThe Null Hypothesis

• The hypothesis that our subjects came The hypothesis that our subjects came from a population of normal responders.from a population of normal responders.

• The hypothesis that watching a violent The hypothesis that watching a violent video does not change mean number of video does not change mean number of aggressive interpretations.aggressive interpretations.

• The hypothesis we usually want to The hypothesis we usually want to reject.reject.

Important ConceptsImportant Concepts

• Concepts critical to hypothesis Concepts critical to hypothesis testingtesting DecisionDecision

Type I errorType I error

Type II errorType II error

Critical valuesCritical values

One- and two-tailed testsOne- and two-tailed tests

DecisionsDecisions

• When we test a hypothesis we draw a When we test a hypothesis we draw a conclusion; either correct or incorrect.conclusion; either correct or incorrect. Type I errorType I error

• Reject the null hypothesis when it is Reject the null hypothesis when it is actually correct.actually correct.

Type II errorType II error• Retain the null hypothesis when it is Retain the null hypothesis when it is

actually false.actually false.

Type I ErrorsType I Errors

• Assume violent videos really have no Assume violent videos really have no effect on associationseffect on associations

• Assume we conclude that they do.Assume we conclude that they do.

• This is a Type I errorThis is a Type I error Probability set at alpha (Probability set at alpha ())

usually at .05usually at .05

Therefore, probability of Type I error = .05Therefore, probability of Type I error = .05

Type II ErrorsType II Errors• Assume violent videos make a differenceAssume violent videos make a difference

• Assume that we conclude they don’tAssume that we conclude they don’t

• This is also an error (Type II)This is also an error (Type II) Probability denoted beta (Probability denoted beta ())

• We can’t set beta easily.We can’t set beta easily.• We’ll talk about this issue later.We’ll talk about this issue later.

• Power = (1 - Power = (1 - ) = probability of correctly ) = probability of correctly rejecting false null hypothesis. rejecting false null hypothesis.

Critical ValuesCritical Values• These represent the point at which we These represent the point at which we

decide to reject null hypothesis.decide to reject null hypothesis.

• e.g. We might decide to reject null when e.g. We might decide to reject null when ((pp|null) |null) << .05. .05. Our test statistic has some value with Our test statistic has some value with pp = .05 = .05

We reject when we exceed that value.We reject when we exceed that value.

That value is the critical value.That value is the critical value.

One- and Two-Tailed TestsOne- and Two-Tailed Tests

• Two-tailed test rejects null when Two-tailed test rejects null when obtained value too extreme in obtained value too extreme in eithereither directiondirection Decide on this before collecting data.Decide on this before collecting data.

• One-tailed test rejects null if obtained One-tailed test rejects null if obtained value is too low (or too high)value is too low (or too high) We only set aside one direction for We only set aside one direction for

rejection.rejection.

Cont.

One- & Two-Tailed One- & Two-Tailed ExampleExample

• One-tailed testOne-tailed test Reject null if violent video group had too many Reject null if violent video group had too many

aggressive associatesaggressive associates• Probably wouldn’t expect “too few,” and Probably wouldn’t expect “too few,” and

therefore no point guarding against it.therefore no point guarding against it.

• Two-tailed testTwo-tailed test Reject null if violent video group had an Reject null if violent video group had an

extreme number of aggressive associates; extreme number of aggressive associates; either too many or too few.either too many or too few.

Testing Hypotheses: Testing Hypotheses: knownknown

• HH00: : = 5.65 = 5.65

• HH11: : 5.65 5.65(Two-tailed)(Two-tailed)

• Calculate Calculate p p (sample mean) = 7.10 if (sample mean) = 7.10 if = 5.65 = 5.65

• Use Use zz from normal distribution from normal distribution

• Sampling distribution would be normalSampling distribution would be normal

Using z Using z To Test To Test HH00 • Calculate Calculate zz

• If If zz > > ++ 1.96, reject 1.96, reject H H00

• 3.22 > 1.96 3.22 > 1.96 The difference is significant.The difference is significant.

22.345.45.1

1005.4

65.51.7

n

Xz

Cont.

z--cont.z--cont.

• Compare computed Compare computed zz to histogram of to histogram of sampling distributionsampling distribution

• The results should look consistent.The results should look consistent.

• Logic of testLogic of test Calculate probability of getting this mean Calculate probability of getting this mean

if null true.if null true.

Reject if that probability is too small.Reject if that probability is too small.

Testing When Testing When Not Not KnownKnown

• Assume same example, but Assume same example, but not not knownknown

• Can’t substitute Can’t substitute ss for for because because ss more likely to be too smallmore likely to be too small See next slide.See next slide.

• Do it anyway, but call answer Do it anyway, but call answer tt

• Compare Compare t t to tabled values of to tabled values of tt..

Sampling Distribution of Sampling Distribution of the Variancethe Variance

Sample variance

800.0750.0

700.0650.0

600.0550.0

500.0450.0

400.0350.0

300.0250.0

200.0150.0

100.050.0

0.0

Fre

qu

en

cy

1400

1200

1000

800

600

400

200

0

138.89

Population variance = 138.89

n = 5

10,000 samples

58.94% < 138.89

tt Test for One Mean Test for One Mean

• Same as Same as zz except for except for ss in place of in place of . .

• For Bushman, For Bushman, ss = 4.40 = 4.40

30.344.

45.1

100

40.465.51.7

n

sX

t

Degrees of FreedomDegrees of Freedom• Skewness of sampling distribution of Skewness of sampling distribution of

variance decreases as variance decreases as nn increases increases

• tt will differ from will differ from zz less as sample size less as sample size increasesincreases

• Therefore need to adjust Therefore need to adjust tt accordingly accordingly

• dfdf = = nn - 1 - 1

• tt based on based on dfdf

t t DistributionDistribution

Two-Tailed Significance Level

df .10 .05 .02 .0110 1.812 2.228 2.764 3.16915 1.753 2.131 2.602 2.94720 1.725 2.086 2.528 2.84525 1.708 2.060 2.485 2.78730 1.697 2.042 2.457 2.750

100 1.660 1.984 2.364 2.626

ConclusionsConclusions

• With With nn = 100, = 100, tt.05.0599 = 1.9899 = 1.98

• Because Because tt = 3.30 > 1.98, reject = 3.30 > 1.98, reject HH00

• Conclude that viewing violent video Conclude that viewing violent video leads to more aggressive free leads to more aggressive free associates than normal.associates than normal.

Factors Affecting Factors Affecting tt

• Difference between sample and Difference between sample and population meanspopulation means

• Magnitude of sample varianceMagnitude of sample variance

• Sample sizeSample size

Factors Affecting DecisionFactors Affecting Decision

• Significance level Significance level

• One-tailed versus two-tailed testOne-tailed versus two-tailed test

Size of the EffectSize of the Effect• We know that the difference is We know that the difference is

significant.significant. That doesn’t mean that it is important.That doesn’t mean that it is important.

• Population mean = 5.65, Sample Population mean = 5.65, Sample mean = 7.10mean = 7.10

• Difference is nearly 1.5 words, or Difference is nearly 1.5 words, or 25% more violent words than normal.25% more violent words than normal.

Cont.

Effect Size (cont.)Effect Size (cont.)

• Later we will express this in terms of Later we will express this in terms of standard deviations.standard deviations. 1.45 units is 1.45/4.40 = 1/3 of a 1.45 units is 1.45/4.40 = 1/3 of a

standard deviation.standard deviation.

Confidence Limits on Confidence Limits on MeanMean

• Sample mean is a point estimateSample mean is a point estimate

• We want interval estimateWe want interval estimate Probability that interval computed this Probability that interval computed this

way includes way includes = 0.95 = 0.95

XstXCI 025.95.

For Our DataFor Our Data

97.723.687.01.7

44.098.11.7025.95.

XstXCI

Confidence IntervalConfidence Interval• The interval does not include 5.65--the The interval does not include 5.65--the

population mean without a violent videopopulation mean without a violent video

• Consistent with result of Consistent with result of tt test. test.

• Confidence interval and effect size tell us Confidence interval and effect size tell us about the magnitude of the effect.about the magnitude of the effect.

• What can we conclude from confidence What can we conclude from confidence interval?interval?