# Random samples of square-kilometer plots were taken in different

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Where are the deer? Random samples of square-kilometer plots were taken in different ecological locations of Mesa Verde National Park. The deer counts per square kilometer were recorded and are shown in the following table.

Mountain Brush

Sagebrush Grassland

Pinon Juniper

32

19

9

29

60

4

19

16

9

28

19

11

Shall we reject or accept the claim that there is no difference in the mean number of deer per square kilometer in these different ecological locations? Use a 1% level of significance.

(a) What is the level of significance?

State the null and alternate hypotheses.

Ho: μ1 = μ2 = μ3; H1: Exactly two means are equal. Ho: μ1 = μ2 = μ3; H1: At least two means are equal. Ho: μ1 = μ2 = μ3; H1: All three means are different. Ho: μ1 = μ2 = μ3; H1: Not all the means are equal.

(b) Find SSTOT, SSBET, and SSW and check that SSTOT = SSBET + SSW. (Use 3 decimal places.)

SSTOT

=

SSBET

=

SSW

=

Find d.f.BET, d.f.W, MSBET, and MSW. (Use 2 decimal places for MSBET, and MSW.)

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. (Use 4 decimal places.)

(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 = 0.01, we do not reject H0. Since the P value is less than or equal to the level of significance at α = 0.01, we reject H0. Since the P value is greater than the level of significance at α = 0.01, we reject H0. Since the P value is less than or equal to the level of significance at α = 0.01, we do not reject H0.

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

At the 1% level of significance there is insufficient evidence to conclude that the means are not all equal. At the 1% level of significance there is sufficient evidence to conclude that the means are not all equal. At the 1% level of significance there is insufficient evidence to conclude that the means are all equal. At the 1% level of significance there is sufficient evidence to conclude that the means are 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— Do not reject H0. Reject H0.

Within groups

Total

.

.

.

.

.

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Wild irises are beautiful flowers found throughout the United States, Canada, and northern Europe. This problem concerns the length of the sepal (leaf-like part covering the flower) of different species of wild iris. Data are based on information taken from an article by R. A. Fisher in Annals of Eugenics (Vol. 7, part 2, pp. 179 -188). Measurements of sepal length in centimeters from random samples of Iris setosa (I), Iris versicolor (II), and Iris virginica (III) are as follows below.

I

II

III

5.4

5.4

6.3

4.2

6.3

5.7

5.5

6.2

4.6

5.8

4.3

7.5

4.4

5.2

5.2

5.2

6.2

6.1

5.7

5.5

6.7

Shall we reject or not reject the claim that there are no differences among the population means of sepal length for the different species of iris? Use a 1% level of significance.

(a) What is the level of significance?

State the null and alternate hypotheses.

Ho: μ1 = μ2 = μ3; H1: All three means are different. Ho: μ1 = μ2 = μ3; H1: Not all the means are equal. Ho: μ1 = μ2 = μ3; H1: At least two means are equal. Ho: μ1 = μ2 = μ3; H1: Exactly two means are equal.

(b) Find SSTOT, SSBET, and SSW and check that SSTOT = SSBET + SSW. (Use 3 decimal places.)

SSTOT

=

SSBET

=

SSW

=

Find d.f.BET, d.f.W, MSBET, and MSW. (Use 4 decimal places for MSBET, and MSW.)

dfBET

=

dfW

=

MSBET

=

MSW

=

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

What are the degrees of freedom?

(numerator)

(denominator)

(c) Find the P-value of the sample test statistic. (Use 4 decimal places.)

(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 = 0.01, we do not reject H0. Since the P value is less than or equal to the level of significance at α = 0.01, we reject H0. Since the P value is greater than the level of significance at α = 0.01, we reject H0. Since the P value is less than or equal to the level of significance at α = 0.01, we do not reject H0.

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

At the 1% level of significance there is insufficient evidence to conclude that the means are not all equal. At the 1% level of significance there is sufficient evidence to conclude that the means are all equal. At the 1% level of significance there is insufficient evidence to conclude that the means are all equal. At the 1% 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— Do not reject H0. Reject H0.

Within groups

Total

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