Question

M14 #13: There is an archaeological study area located in southwestern New Mexico. Potsherds are broken pieces of prehistoric Native American clay vessels. One type of painted ceramic vessel is called Mimbres classic black-on-white. At three different sites, the number of such sherds was counted in local dwelling excavations.

Site I	Site II	Site III
64	20	15
31	18	37
29	52	65
18	64	27
83		11
58		14
23

A)  Find SS_TOT, SS_BET, and SS_W and check that SS_TOT = SS_BET + SS_W. (Use 3 decimal places.)

B) Find d.f._BET, d.f._W, MS_BET, and MS_W. (Use 4 decimal places for MS_BET, and MS_W.)

C) Find the value of the sample F statistic. (Use 2 decimal places.)

Answer 1

	Site I	Site II	Site III
count, ni =	7	4	6
mean , x̅ i =	43.71428571	38.50	28.17
std. dev., si =	24.58803426	23.0578981	20.51747223
sample variances, si^2 =	604.5714286	531.666667	420.9666667
total sum	306	154	169		629	(grand sum)
grand mean , x̅̅ =	Σni*x̅i/Σni =				37.00
square of deviation of sample mean from grand mean,( x̅ - x̅̅)²	45.08163265	2.25	78.02777778
					TOTAL
SS(between)= SSB = Σn( x̅ - x̅̅)² =	315.5714286	9.00	468.1666667		792.7381
SS(within ) = SSW = Σ(n-1)s² =	3627.428571	1595.00	2104.833333		7327.262

A)

SSbet = 792.738

SSW=7327.262

SST=Σ( x - x̅̅ )² =    7911.750

SSbet + SSW = 792.738 + 7327.262 = 7911.750

so,  SS_TOT = SS_BET + SS_W proved,

B)

no. of treatment , k =   3
df between = k-1 =    2
N = Σn =   17
df within = N-k =   14

mean square between groups , MSB = SSB/k-1 =    396.3690
  
mean square within groups , MSW = SSW/N-k =    523.3759

C)

F-stat = MSB/MSW =    0.76

anova table
	SS	df	MS	F
Between:	792.738	2	396.3690	0.76
Within:	7327.262	14	523.3759
Total:	8120.000	16