The Analysis Tasks
The questions you need to answer in your assignment submission are given below.
Q1. The arborists start with an exploratory analysis on the Diameter at Breast Height (DBH) of the Karri tree data collected.
1.a. Calculate the sample mean and sample standard deviation of the DBH (Tree.Diameter) of the trees in your sample.
1.b. Produce a histogram for the Tree.Diameter measurements. Include this histogram in your submitted assignment properly labelled.
1.c. Comment on the shape (skewness/symmetry) of your histogram from Part 1b.
1.d. A common technique that can be used to remove skewness in data is known as a logtransformation. That is, for each value in your data (denoted by xi), you can log-transform it as yi = log(xi). The function in RStudio that performs a log-transformation on a value is log(). Produce a histogram for the log(Tree.Diameter) measurements. Include this new histogram in your submitted assignment properly labelled.
1.e. Do you think this log-transformation reduced any skewness identified in Part 1c? Explain briefly.
The arborists now explore the leaf data they collected.
1.f. Produce comparative boxplots for Leaf.Length against Leaf.Type. Include this plot in your submitted assignment, properly labelled.
1.g. Describe any differences or similarities in the distribution of the length of leaves among newer (’Youth’) and older (’Adult’) leaves using your comparative boxplots from Part 1f. Include in your answer comments on shape, location, spread and outliers. Your answer should be written in plain English so that others not looking at your boxplots can understand the distribution of these leaf lengths.
Q2. The Australian GeoZight Magazine published an article which claims that the average height of Karri trees is 55m. However, the arborists believe that the true average height is actually lower than 55m and ask you to investigate their claim.
Let µ be the true mean height of Karri trees, which are endemic to the southern region of Western Australia.
2.a. Produce a normal quantile plot of your sample of Tree.Height values (see Section R2.6 “How to produce a normal quantile plot using RStudio”). Include this plot in your submitted assignment, properly labelled.
2.b. By referring to the normal quantile plot obtained in Part 2a, briefly discuss if the Karri tree heights are approximately normally distributed.
2.c. Carry out an appropriate hypothesis test by completing the following steps.
i. State the null (H0) and alternative hypotheses (Ha) relevant to the research objective.
Write the hypotheses out in plain English and mathematically.
ii. State the expression for a suitable test statistic.
iii. Calculate the observed value of the test statistic using your sample.
iv. Write down the sampling distribution for the test statistic when the null hypothesis, H0, is true.
v. Calculate the P-value for this test, showing some working.
vi. Explain what the P-value you obtained in Part 2(c)v calculates, by describing this probability in words and referring to what you observe in your sample.
vii. Write a conclusion to your test in plain English (that is, with no technical mathematical or statistical language, and referring to the scenario given).
2.d. Some assumptions need to be satisfied for the sampling distribution of the test statistic (as given in Part 2c) to be valid. State these assumptions, and briefly discuss whether these assumptions are satisfied.
2.e. Produce a 95% confidence interval for µ, the true mean Karri tree height. For this question you may assume that it is appropriate to use a t-distribution. Write down all the required steps to calculate this interval.
2.f. Interpret your confidence interval (constructed in Part 2e) in plain English (that is, with no technical mathematical or statistical language, and referring to the scenario given).
2.g. Explain whether your confidence interval (constructed in Part 2e) is consistent with the results of your hypothesis test in Part 2c.
2.h. Like all realised confidence intervals, your confidence interval (constructed in Part 2e) is of the form: estimate ± margin of error. Does the margin of error you calculated include errors due to measurement error (e.g. if the arborist measuring the tree height sometimes wrote down a slightly inaccurate value)? Briefly justify your answer.
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