6. (One-way classified data) Three pottery shards from four widely scattered and

Question

6. (One-way classified data) Three pottery shards from four widely scattered and now extinct Native American tribes have been collected by a museum. Archeologists were asked to estimate the age of the shards. Their estimates are for various tribes are summarized in the following table Lakeside Deep Gorge| Willow Ridge Azalea Hill 1200 800 950 850 900 1100 1800 1450 1150 950 1200 1150 Based on the results shown in this table, is it conceivable that the four tribes were contemporaries of one another? Use a-level F-test, with -0.05 Hint: fas(0.05) = 4.1

Explanation / Answer

From the data the following have been calculated:

The Hypothesis:

H0: The 4 tribes were contemporaries of each other, i.e the mean age of the shards are the same for the 4 sites.

Ha: The 4 tribes were not contemporaries of each other, i.e the mean age of the shards are different for at least 2 of the sites.

The Test Statitic:

The table below shows the calculation of F. (I have also done the calculations for the table at the end)

Fobserved = 7.898

F crtical3,8 (0.05) = 4.1

The Decision rule is if Fobserved is > F critical, Reject H0.

The Decision: Since Fobserved (7.898) is > F critical (4.1), We Reject H0.

The Conclusion: There is sufficient evidence at the 95% significance level to conclude that the 4 tribes were not contemporaries of one another.

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Calculations for the ANOVA table.

The overall mean = (983.3 + 983.3 + 1467 + 1100)/4 = 1133.4

SS treatment = SUM n* ( xi - overall mean)2 = 3 * (983.3 - 1133.4)2 + 3 * (983.3 - 1133.4)2 + 3 * (1467 - 1133.4)2 + 3 * (1100 - 1133.4)2 = 472393.62

df1 = k - 1 = 4 - 1 = 3

MSTR = SS treatment/df1 = 472393.62 / 3 = 157464.54

SSerror = SUM [(n - 1) * Variance] = 2 * 40833 + 2 * 10833 + 2 * 10583.33 + 2 * 17500 = 159498.66

df2 = N - k = 12 - 4 = 8

Therefore MS error = SSerror/df2 = 159498.66/8= 19937.63

F = MSTR/MSE = 157464.54/19937.66 = 7.898

Lakeside Deep Gorge Willow Ridge Azlea Hill Total Mean(x bar) 983.3 983.3 1467 1100 SD 202.07177 104.081699 102.8753129 132.2875656 0 0 Variance 40833.0000 10833.0000 10583.3300 17500.0000 0.0000 0 n 3 3 3 3 0 12 columns 4

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