Transformations and Confidence Intervals
An ecologist has collected data on the height (in cm) of the shrub Melicytus micranthu growing in an area where attempts have been made to control the noxious vine old man's beard. This is the data
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Height (cm)
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56
|
|
67
|
|
135
|
|
72
|
|
104
|
|
72
|
|
80
|
|
120
|
|
38
|
|
64
|
|
115
|
|
29
|
|
56
|
|
72
|
|
46
|
|
44
|
|
50
|
|
225
|
|
70
|
|
46
|
|
29
|
|
62
|
|
135
|
|
44
|
|
29
|
|
34
|
|
72
|
|
59
|
|
93
|
|
62
|
|
54
|
|
93
|
|
56
|
|
36
|
|
48
|
|
70
|
|
67
|
|
72
|
|
56
|
|
72
|
|
96
|
|
146
|
|
80
|
|
82
|
|
120
|
|
92
|
|
36
|
|
53
|
|
38
|
|
44
|
|
26
|
1. Exploratory data analysis
a) Construct a histogram of the heights. (Minitab hint: Graph: Histograms: Simple)
b) Comment of the shape of the distribution of heights.
c) Find the mean and median height of the shrubs, and comment on the difference between them.
2. Confidence interval
a) Construct and interpret a 95% confidence interval for the mean height of Melicytus micranthu in this area.
3. Transforming the data
a) Create 3 new columns in Minitab containing the following transformations of the shrub heights:
i. square root,
ii. natural log and
iii. inverse (i.e. 1/value)
b) Construct a histogram of the transformed heights for each of the three transformations.
c) Which transformation makes the data appear approximately Normally Distributed?
4. Using the transformation you chose in 2c):
a) Find the mean and median of the transformed heights.
b) Convert the mean and median of the transformed data back to the original units.
c) Discuss whether the back-transformed mean is closer to the mean or median of the original data, and why.
It is known that in areas where the noxious vine old man's beard has never grown (natural area) the mean height of the shrub Melicytus micranthu grows to an average height of 82.5cm.
5. Using the untransformed data, conduct a hypothesis test to determine if the average height of the shrub in the controlled area is significantly smaller than the average height of the shrub in the natural area.
(a) State your null and alternative hypotheses in words and symbols.
(b) Use Minitab to calculate the test statistic and corresponding p-value.
(c) Explain whether you have evidence for or against the null hypothesis
(d) State your conclusion in a form that a non-statistician would understand.