Aresearcherisinterestedincomparingtheperformanceandaptitudesoflawstudents for statisticalmethods,depending ontheclustertheyareregisteredin.

Since there are more than two clusters in the program, she decides to test different hypothesesbycomparingthemeanfinalgrade(outof100points)obtainedbystudentsfromagivenpairofclusters,usingdatafromFall2000asasample.Tosimplifymatters, she renamedtheclusters #1,#2,#3,etc.inthedataforconfidentialitypurposes.

Shelooksfirstatthedescriptivestatisticsforthegradesofpeopleinclusters#1and#2:

Cluster1 Cluster2

Mean	79.0289323	Mean	77.4
StandardError	2.375690812	StandardError	2.352617525
Median	79	Median	78
Mode	79	Mode	81
StandardDeviation	9.472778844	StandardDeviation	8.82435044
SampleVariance	89.73353903	SampleVariance	77.86916069
Kurtosis	1.825855777	Kurtosis	2.370477133
Skewness	-1.020508722	Skewness	-1.051488911
Range	38	Range	6
Minimum	52	Minimum	51
Maximum	94	Maximum	96
Sum	1172	Sum	1122

Count 15 Count 15

Giventhemeansforeachcluster, theresearcherstates thefollowinghypotheses:

H0:thereisnodifferencebetweenthemeansofpeople fromcluster1andpeople fromcluster2 H1: the mean for people in cluster 1 is higher than the mean for people in cluster 2

1. Whatversionofat-test(one-tailedortwo-tailed)wouldbemostappropriatetotrytofind support for this research hypothesis? Briefly state why.

Theresultsforthet-testareasfollows:
	Cluster1	Cluster2
Mean	79.0289323	77.4
Variance	89.73353903	77.86916069
Observations	15	15
PooledVariance	83.87619048
HypothesizedMeanDifference	0
df	28
tStat	0.498379342
P(T<=t)one-tail	0.311054567
tCriticalone-tail	1.701130934
P(T<=t)two-tail	0.622109134

tCriticaltwo-tail 2.048407142

2. Lookingattheresults, canshereject thenullhypothesis?Be specific.

Nowconsiderthefollowingresults fromatwo-samplet-testconductedtocomparethemeans forclusters#3and#4.

	Cluster3	Cluster4
Mean	72.53333333	77.8
Variance	78.40952381	96.31428571
Observations	15	15
PooledVariance	87.36190476
HypothesizedMeanDifference	0
df	28
tStat	-1.543
P(T<=t)one-tail	0.067
tCriticalone-tail	1.701
P(T<=t)two-tail	0.134
tCriticaltwo-tail	2.048

3. Supposethattheresearcherisusingatwo-tailedtest.Whatisthecriteria to"pass"thetest(i.e., to reject the null hypothesis)?

4. Atthelevelofsignificanceused(=0.05), isthedifferencebetweenthegroups'means statistically significant? Justify.

5. Atthelevelofsignificanceused(=0.05),isthemeanforthe#4groupstatisticallysignificantly larger than the mean for the #3 group? Justify.