Question

7.31 Tree diameter confidence interval. A study of 584 longleaf pine trees in the Wade Tract in Thomas County, Georgia, is described in Example 6.1 (page 352). For each tree in the tract, the researchers measured the diameter at breast height (DBH). This is the diameter of the tree at a height of 4.5 feet, and the units are centimeters (cm). Only trees with DBH greater than 1.5 cm were sampled. Here are the diameters of a random sample of 40 of these trees: PINES 0 10.5 47.2 4.3 43.6 13.3 11.4 7.8 2.3 26.0 2.7 38.1 44.6 18.3 69.3 2.2 31.5 52. 2 44.4 11.4 40.3 9.2 16.9 51.5 22.3 26.1 35.7 4.9 43.3 17.6 5.4 39.7 37.5 40.5 44.2 32.6 29.1 31.8 2.2 51.8 27.9 (a) Use a histogram or stemplot and a boxplot to examine the distribution of DBHs. Include a Normal quantile plot if you have the necessary software. Write a careful description of the distribution. (b) Is it appropriate to use the methods of this section to find a 95% confidence interval for the mean DBH of all trees in the Wade Tract? Explain why or why not. (c) Report the mean with the margin of error and the confidence interval. Write a short summary describing the meaning of the confidence interval. (d) Do you think these results would apply to other simi- lar trees in the same area? Give reasons for your answer.
please answer abc

show your work

figure 6.1

Answer 1

10.5	13.3	26	18.3	52.2	9.2	26.1	17.6	40.5	31.8
47.2	11.4	2.7	69.3	44.4	16.9	35.7	5.4	44.2	2.2
4.3	7.8	38.1	2.2	11.4	51.5	4.9	39.7	32.6	51.8
43.6	2.3	44.6	31.5	40.3	22.3	43.3	37.5	29.1	27.9

A -

Boxplot

Sample size: 40
Median: 28.5
Minimum: 2.2
Maximum: 69.3
First quartile: 10.725
Third quartile: 42.6
Interquartile Range: 31.875
Outliers: none

Histogram

Frequency Table
DBH	Count
0-11.9	12
12-23.9	5
24-35.9	8
36-47.9	11
48-59.9	3
60-71.9	1

0 12 24 36 486072

As seen in the boxplot and histogram plot, we have more values clustered at the right side of the distribution. Hence, this is a right (or positive skewed data)

B -

As the sample is not close to normal distribution, the population can not be approximated using the confidence intervals because the confidence intervals are based on the assumptions that the population and sample follows a normal distribution which is violated here.

C -

Mean of the sample (point estimate) = 27.3

Standard deviation of the sample = 17.7

t value (for (40-1) degrees of freedom and 0.05 significance level for a two tailed hypothesis) = 2

Standard error = std dev / sqrt (sample size - 1)

= 2.83

Margin of error = t vale * std error

= 5.7

Confidence interval -

27.3 - 5.7 to 27.3 + 5.7

= 21.6 to 33