1.A study of 584 longleaf pine trees in the Wade Tract in Thomas County, Georgia
ID: 3232243 • Letter: 1
Question
1.A study of 584 longleaf pine trees in the Wade Tract in Thomas County, Georgia had several purposes. To see if there is a difference in their sizes (in diameters) in two separate areas of the Wade Tract (northern and southern areas), a random sample of 30 trees from the northern area and 30 from the southern area was taken and are as follows:
diameter=c(27.8,14.5,39.1,3.2,58.8,55.5,25,5.4,19,30.6,15.1, 3.6,28.4,15,2.2,14.2,44.2,25.7,11.2,46.8,36.9,54.1, 10.2,2.5,13.8,43.5,13.8,39.7,6.4,4.8,44.4,26.1,50.4, 23.3,39.5,51,48.1,47.2,40.3,37.4,36.8,21.7,35.7,32, 40.4,12.8,5.6,44.3,52.9,38,2.6,44.6,45.5,29.1,18.7, 7,43.8,28.3,36.9,51.6) direction=c(rep('North',each=30),rep('South',each=30)) trees=data.frame(diameter,direction)
(a) Estimate the true difference in mean tree sizes between the northern and southern parts of the Wade Tract with 95% confidence. Interpret.
(b) Is there a significant difference in the mean diameter of trees in the north versus the trees in the south? Conduct hypothesis test.
(c) State the kind of error could have been made in context of the problem.
(d) Now do part b again in R
The summary statistics are as follows: ni x s ¯ i North 30 23.7 17.5 South 30 34.53 14.258
Explanation / Answer
a)
direction N Mean StDev SE Mean
North 30 23.7 17.5 3.2
South 30 34.5 14.3 2.6
The true difference in mean tree sizes between the northern and southern parts of the Wade Tract is -10.83.
and its 95% confidence interval is (-19.09, -2.57)
The population mean difference of north and souther tres sizes is lies between (-19.09, -2.57) with 95 % confidence.
b) T-Test of difference = 0 (vs not =): T-Value = -2.63 P-Value = 0.011 DF = 55.
The estimated p-value is 0.011. Hence, we can not accept the null hypothesis and conclude that there is a significant difference in the mean diameter of trees in the north versus the trees in the south at 0.05 level of significance.
(c) State the kind of error could have been made in context of the problem.
Ans: If the two samples are not follow the normal distribution this test will have an error. Because the t-test is done under the assumption of normality.
(d) Now do part b again in R
> t.test(North, South)
Welch Two Sample t-test
data: North and South
t = -2.6286, df = 55.725, p-value = 0.01106
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
-19.090199 -2.576468
sample estimates:
mean of x mean of y
23.70000 34.53333
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