Question

A sample of 10 longleaf pine trees each are taken from the western and eastern halves of the Wade Tract in Thomas County, Georgia. The diameters for the trees are given:

Western	17.2	44.6	44.1	35.5	51	21.6	44.1	11.2	36	42.1
Eastern	23.5	43.5	6.6	11.5	17.2	38.7	2.3	31.5	10.5	23.7

(a) Is there a difference between the mean diameters of the trees in the two halves? Perform a test of significance at level α = .05 based on Satterthwaite’s approximation (without assuming equality of popu- lation standard deviations). Carefully show all steps of your test.
(b) Now, assume that the population standard deviations are equal. Perform a test of significance at levelα = .05. Carefully show all steps of your test.

Answer 1

For Western

= 34.74, s1 = 13.45, n1 = 10

For Eastern

= 20.9, s2 = 13.81, n2 = 10

a) H₀:

    H₁:

The test statistic t = ()/sqrt(s1^2/n1 + s2^2/n2)

                             = (34.74 - 20.9)/sqrt((13.45)^2/10 + (13.81)^2/10)

                             = 2.27

DF = (s1^2/n1 + s2^2/n2)^2/((s1^2/n1)^2/(n1 - 1) + (s2^2/n2)^2/(n2 - 1))

      = ((13.45)^2/10 + (13.81)^2/10)^2/(((13.45)^2/10)^2/9 + ((13.81)^2/10)^2/9)

      = 18

At = 0.05, the critical vallue is t_{0.025, 18} = +/- 2.101

Since the test statistic value is greater than the upper critical value (2.27 > 2.101), we should reject the null hypothesis.

So at = 0.05, we can conclude that there is a difference between the mean diameters of the trees in the two halves.

b) H₀:

    H₁:

s_p² = ((n1 - 1)s1^2 + (n2 - 1)s2^2)/(n1 + n2 -2)

      = (9 * (13.45)^2 + 9 * (13.81)^2)/(10 + 10 - 2)

      = 185.8093

The test statistic t = ()/sqrt(s_p²/n1 + s_p²/n2)

                             = (34.74 - 20.9)/sqrt(185.8093/10 + 185.8093/10)

                            = 2.27

DF = 10 + 10 - 2 = 18

At = 0.05, the critical value is t_{0.025, 18} = +/- 2.101

Since the test statistic value is greater than the upper critical value (2.27 > 2.101), so we should reject the null hypothesis.

So at = 0.05, we can conclude that there is a difference between the mean diameters of the trees in the two halves.