Question

Another type of painted ceramic vessel is called three-circle red-on-white ( Mimbres Mogollon Archaeology). At four different sites in an archaeological region, the number of such sherds was counted in local dwelling excavations.

Site I	Site II	Site III	Site IV
18	11	37	17
28	4	19	23
9	35	11	17
22	3	45	36
10	29		19
	11		11

Shall we reject or not reject the claim that there is no difference in the population mean three-circle red-on-white sherd counts for the four sites? Use a 5% level of significance.

(a) What is the level of significance?


State the null and alternate hypotheses.

H_o: μ₁ = μ₂ = μ₃ = μ₄; H₁: Exactly three means are equal.H_o: μ₁ = μ₂ = μ₃ = μ₄; H₁: Exactly two means are equal.    H_o: μ₁ = μ₂ = μ₃ = μ₄; H₁: All four means are different.H_o: μ₁ = μ₂ = μ₃ = μ₄; H₁: Not all the means are equal.

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

SSTOT	=
SSBET	=
SSW	=

Find d.f._BET, d.f._W, MS_BET, and MS_W. (Use 3 decimal places for MS_BET, and MS_W.)

dfBET	=
dfW	=
MSBET	=
MSW	=

Find the value of the sample F statistic. (Use 3 decimal places.)


What are the degrees of freedom?
(numerator)
(denominator)

(c) Find the P-value of the sample test statistic.

P-value > 0.1000.050 < P-value < 0.100    0.025 < P-value < 0.0500.010 < P-value < 0.0250.001 < P-value < 0.010P-value < 0.001

(d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis?

Since the P-value is greater than the level of significance at α = 0.05, we do not reject H₀.Since the P-value is less than or equal to the level of significance at α = 0.05, we reject H₀.    Since the P-value is greater than the level of significance at α = 0.05, we reject H₀.Since the P-value is less than or equal to the level of significance at α = 0.05, we do not reject H₀.

(e) Interpret your conclusion in the context of the application.

At the 5% level of significance there is insufficient evidence to conclude that the means are not all equal.At the 5% level of significance there is sufficient evidence to conclude that the means are all equal.    At the 5% level of significance there is insufficient evidence to conclude that the means are all equal.At the 5% level of significance there is sufficient evidence to conclude that the means are not all equal.

(f) Make a summary table for your ANOVA test.

Source of Variation	Sum of Squares	Degrees of Freedom	MS	F Ratio	P Value	Test Decision
Between groups					---Select--- p-value > 0.100 0.050 < p-value < 0.100 0.025 < p-value < 0.050 0.010 < p-value < 0.025 0.001 < p-value < 0.010 p-value < 0.001	---Select--- Do not reject H0. Reject H0.
Within groups
Total

Answer 1

a) level of significance =.05

H_o: μ₁ = μ₂ = μ₃ = μ₄; H₁: Not all the means are equal.

b)   SS_TOT=2665.81 SS_BET=411.61, SS_W=2254.20 and checked that SS_TOT = SS_BET + SS_W

c) d.f._BET=3,  d.f._W=17, MS_BET=137.2 , MS_W=132.6

Sample F statistic=1.035

df(numerator)=3
df(denominator)=17

c) P-value > 0.1000

d)  Since the P-value is greater than the level of significance at α = 0.05, we do not reject H₀

e)  At the 5% level of significance there is insufficient evidence to conclude that the means are not all equal

f) ANOVA Table

Df SS MS   F    p value Decision
Between 3 411.61 137.2 1.0347 p-value > 0.100   Do not reject H0
Within 17 2254.20 132.6

Total 20 2665.81