8 - One Sample t

8/8/2019 8 - One Sample t

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HomeworkHomework

Set IISet II

12, 13, 1412, 13, 14

NOTE: For questions 13 and 14, you will needNOTE: For questions 13 and 14, you will needto calculate boththe sample mean andto calculate boththe sample mean andvariance.variance.


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BackgroundBackground

tt tests allow us to test differencestests allow us to test differencesbetween populations when we dontknowbetween populations when we dontknowall the information.all the information.

Stat developed to test for differences inStat developed to test for differences inGuinness.Guinness. (comparing one batch to another)(comparing one batch to another)

tt distributions are slightly different fromdistributions are slightly different fromthose weve been using.those weve been using.


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OneOne--SampleSample tt TestTest

Used whenUsed when MMpoppop is known, butthe variance isis known, butthe variance isnotnot..

Assumption 1Assumption 1 DV is normally distributedDV is normally distributed

NN>30, should meetthis.>30, should meetthis.

Fairly accurate even when this assumption isFairly accurate even when this assumption is

violatedviolated

Assumption 2Assumption 2

a)randomsamplea)randomsample

b) scores are independentb) scores are independent


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Steps for OneSteps for One--SampleSample tt TestTest

Similartohypothesis testing withSimilartohypothesis testing with zz scores.scores.

Step 1: ID populations and hypotheses.Step 1: ID populations and hypotheses.

Step 2: Determine characteristics ofStep 2: Determine characteristics ofcomparison distribution.comparison distribution.

MM is known. Justplug in.is known. Justplug in.

SESE ?!?!?!?!


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Estimating PopulationEstimating Population

VarianceVariance If sample is random, sample varianceIf sample is random, sample variance

should approximate population variance.should approximate population variance.

However . . .However . . . Sample variance generally smallerthanSample variance generally smallerthan

population (i.e., its a biased estimate)population (i.e., its a biased estimate) So, divide bySo, divide by NN--1 to correct for bias1 to correct for bias

NN--1 is called degrees of freedom1 is called degrees of freedom

The formula:The formula:S

2 !(XM)2

df


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Step 3. CalculateStep 3. Calculate ttcritcrit.. Need toknow:Need toknow:

oneone-- vs. twovs. two--tailed test?tailed test?

pp valuevalue -- .05, .01.05, .01 degrees of freedom (degrees of freedom (dfdf))

Identify correspondingIdentify corresponding tt value on Table Avalue on Table A--

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IF, exactIF, exactdfdf noton table, use the closestnoton table, use the closestnumberthat isnumberthat is belowbelow your actualyour actual dfdf..


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Step 4. DetermineStep 4. Determine ttobtobt..

Similarto calculatingSimilarto calculating zz score:score:

MM=sample mean=sample mean

MMpoppop=population mean=population mean

SSMM=estimated standard error=estimated standard error

t!MMpop

SM


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Step 5. Make a decision regarding theStep 5. Make a decision regarding the HH00..

IfIfttobtobt is more extreme thanis more extreme than ttcritcrit, reject, rejectHH00..


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ExampleExample

Determine whether individuals who are stressedDetermine whether individuals who are stressedconsume more alcohol per sitting compared toconsume more alcohol per sitting compared toindividuals in general.individuals in general.

Six stressed individuals reportconsuming 59, 45,Six stressed individuals reportconsuming 59, 45,56, 57, 67, and 56 ozof alcohol per sitting.56, 57, 67, and 56 ozof alcohol per sitting.

In general, individuals consume aIn general, individuals consume a MM=50.00 oz=50.00 ozper sitting.per sitting.


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Step 1Step 1

Identify populations and hypotheses:Identify populations and hypotheses:

Population 1: StressedPopulation 1: Stressed

Pop2: NonPop2: Non--StressedStressed Null: Pop1 = Pop2Null: Pop1 = Pop2

Alt: Pop. 1 > Pop. 2Alt: Pop. 1 > Pop. 2


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Step 2.Step 2.

Identify sampleIdentify sample MM, and estimate, and estimate SS22 ofofpopulation and distribution ofpopulation and distribution of MM, and, and S.S.

MM=(59+45+56+57+67+56)/6=56.67=(59+45+56+57+67+56)/6=56.67

SS22=SS=SStotaltotal

//dfdf= 249.48/5 = 49.90= 249.48/5 = 49.90

This is the estimated population variance!This is the estimated population variance!


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Step 2 (cont)Step 2 (cont)

Calculate the estimated variance forCalculate the estimated variance fordistribution of means:distribution of means:

SSMM22

=49.90/6 = 8.32=49.90/6 = 8.32

Calculate standard error:Calculate standard error:

Take square rootof 8.32=2.88Take square rootof 8.32=2.88


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Step 3Step 3

DetermineDetermine ttcritcrit..

We have a sample of 6,We have a sample of 6, dfdf=5 (=5 (NN--1)1)

We have oneWe have one--tailed testand settailed testand setp=p=.05.05

ttcritcrit=2.02=2.02


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Steps 4 & 5.Steps 4 & 5.

CalculateCalculate ttobtobt..

(56.67(56.67--50.00)/2.88=2.3250.00)/2.88=2.32

Step 5.Step 5. tt(5)=2.32 ismoreextremethan(5)=2.32 ismoreextremethanttcritcrit=2.02. Wereject=2.02. Wereject

thenull.thenull.

Would we reject if using a twoWould we reject if using a two--tailed test? Whytailed test? Whyor why not?or why not?


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Your Turn . . .Your Turn . . .

Determine whether MOC students differ inDetermine whether MOC students differ inlevels of physical activity compared tolevels of physical activity compared tocollege students in general.college students in general.

A random sample ofA random sample of NN=5 MOC students=5 MOC studentsare selected and report spending 3, 6, 8,are selected and report spending 3, 6, 8,11, and 14 hours per week atthe rec.11, and 14 hours per week atthe rec.

We know in general college studentsWe know in general college studentsspend 12 hours atthe rec per week.spend 12 hours atthe rec per week.


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Important NumbersImportant Numbers


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Important NumbersImportant Numbers

Estimated Population Variance = 18.3Estimated Population Variance = 18.3

Variance of distribution of means = 3.66Variance of distribution of means = 3.66

Estimated standard error = 1.91Estimated standard error = 1.91

ttcritcrit==2.7762.776

ttobtobt==--1.881.88

Fail torejectFail torejectHH00..


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SPSS ExampleSPSS Example

13, 15, 7, 23, 11, 6, 16, 19, 10, 1213, 15, 7, 23, 11, 6, 16, 19, 10, 12

This is yourtestvariable. Enter inThis is yourtestvariable. Enter in COLUMNCOLUMN 11

Goto Analyze, Compare Means, OneGoto Analyze, Compare Means, One--Sample T TestSample T Test

Move testvariable into TestVariable box.Move testvariable into TestVariable box.

Enter TestValue (Enter TestValue (QQ==18) and click OK18) and click OK


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SPSS OuputSPSS Ouput

T-Test

One Sample Statistics

NN MeanMean Std.Std.DeviationDeviation

Std. ErrorStd. ErrorMeanMean

VAR00001VAR00001 1010 13.2013.20 5.245105.24510 1.658651.65865

One SampleTest

TestValue=18TestValue=18

tt dfdf sig (2sig (2--tailed)tailed) Mean Diff.Mean Diff.

VAR00001VAR00001 --2.8942.894 99 .018.018 --4.80004.8000

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8 - One Sample t