(a) [3 marks] Your original dataset (DCE2) contains four variables of interest (ignore rep). Name them and for each variable identify the type of variable, and describe its sample space.
(b) [3 marks] What is your statistical population in this study? Describe, in less than three sentences, how you achieved a random sample from that population.
In this problem you will work with the raw data on leaf widths. W denotes the width of individual leaves.
(a) [2 marks] Compute and report numerical summaries on W (each summary containing min, max, 1st, 2nd and 3rd quartiles + mean) for the north and south sides of the tree.
(b) [2 marks] Produce and report a paired box plot (two boxplots within a single graph) for W
on the south and north sides. Label the y-axis appropriately.
(c) [2 marks] Based on (a) and (b), briefly comment on differences or similarities in terms of centre and spread of the distributions of widths in the north side and compared to the south sides. Provide a biological justification to your findings.
(d) [2 marks] An outlier is an observation that falls outside some pre-determined "fences". An observation smaller than qˆ0.25 − k ׈ IQR or larger than qˆ0.75 + k × IQR, with qˆp denoting the p-sample quantile and IQR = q0.75 − qˆ0.25. Mild (k = 1.5) and strong (k = 3) outliers
fall outside these two "fences". Set k = 1.5 and count the number of outliers for W in both directions beyond these fences. Count the number of mild and strong outliers, for leaves with above- or below-average width, then fill in the 4 cells of a table similar to the one shown below:
Briefly comment on the presence (and potential origin) of outliers in the leaf data with respect to W. Based on your findings, which of the following summaries of {centre, spread} seems to be the most appropriate to summarise width W: {qˆ0.5, IQR} or {x¯, s}?Assuming you have loaded the data and assigned it to an object called DCE2, the following com- mand produces a new data set containing 528 within-student average measurements:
> DCE2.agg <- aggregate( . ~ studentID + side, DCE2, mean)
This command aggregates all the variables in the raw data DCE2 for all the combinations of the factors studentID and side. As a result, each row in the aggregated dataset DCE2.agg reports the average leaf width and length for each side of the tree for each student.
In part (a) you will work with raw and aggregated data. In parts (b)–(e) you will work with the raw leaf width data DCE2.
(a) [3 marks] Compute the sample mean, x¯, and sample standard deviation, s, for the width of leaves using the aggregated data (528 rows) and original data (2731 rows). Plot histograms for the raw leaf widths and for mean leaf widths in the aggregated data. Use the arguments xlim=c(0, 250), breaks=20 for both plots. Briefly compare and contrast the findings from the original and averaged data. Particularly, explain why s in the original data is bigger than the same quantity computed from the averaged data.
(b) [1 mark] Let X be the width of a randomly selected leaf from the South side of the tree. Using the data provide an estimate for p = P(X > 200), i.e. the probability that a randomly selected leaf from the south side of the tree exceeds width of 200 millimetres.
(c) [1 mark] Using the result in (b) compute the (approximate) probability that at least one out of five leaves randomly picked by a student from the south side exceed 200 millimetres width.
(d) [2 marks] Let Y be the number of leaves wider than 200 millimeters out of 5 leaves sampled from the south side of the tree. Give the distribution of Y and specify E(Y) and Var(Y).
(e) [2 marks] Your The lecturer collected a random sample of 5 leaves from the south side of a tree finding that 3 out of 5 leaves are wider then 200 millimetres. Is there enough evidence suggesting that your lecturer collected a sample from a tree different from the one in Data Collection Exercise 2? Justify your answer using an appropriate probability calculation.
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