## The Open University - M248 TMA 01 Question one

**The Open University Statistics TMA 01 Question 1**

(a) A number of Japanese black pine tree seedlings were planted in a rather inaccessible location to which researchers returned at the same time each year in order to measure their growth. The resulting data comprise the height of the young trees (measured on an effectively continuous scale of millimetres) and the age of the trees (1, 2, 3 or 4 years).

(i) Name two graphical displays which are suitable for studying the distribution of the heights of the trees at age 1 year. Give a single reason for the suitability of both displays. [3]

(ii) Unfortunately, the young trees were susceptible to dying off, so the number of trees that remain alive at each age is also a variable of interest. Name a graphical display which is suitable for showing the number of trees alive at each age. Give a reason for your answer. [2]

(iii) Name two graphical displays which are suitable for studying the way that the heights of the trees depend on their ages. Give a separate reason for the suitability of each display. [4]

(b) The Minitab worksheet **snow-depth.mtw** contains measurements of the depth of snow lying at each of n = 114 locations on an Antarctic ice floe in March 2003. The measurements are in centimetres, rounded to

the nearest whole number. The data are in the variable Depth.

(i) Produce Minitab’s default frequency histogram for Depth. Include a copy of this histogram in your answer. Briefly describe the main features of the distribution suggested by this histogram. [3]

(ii) Now use Minitab to produce a frequency histogram for Depth with cutpoints at 0, 10, 20, . . . , 100 cm. Include a copy of this histogram in your answer. Briefly describe the main feature of the

distribution according to this histogram and state why this histogram gives a more clear-cut picture than the default histogram that you obtained in part (b)(i). [4]

(iii) Now **use Minitab** to produce a unit-area histogram for Depth with cutpoints at 0, 10, 20, . . . , 100 cm. Include a copy of this histogram in your answer. In what way(s) does this histogram differ from the frequency histogram that you obtained in part (b)(ii)? The heights of the bars in the histogram that you have just produced should be 0:005 0:016 0:018 0:012 0:011 0:019 0:012 0:004 0:002 0 (each given correct to three decimal places except the last which is exact). Use this information to verify that this histogram does have unit area, as claimed. [5]

(iv) **Using Minitab**, obtain the sample size, sample mean and sample median of the variable Depth and report your result, by copying Minitab text, in the form

Variable N Mean Median

Depth.*.

(where, of course, the asterisks are replaced by numbers).

Comment briefly on the relative size of the sample mean and the

sample median, relating your comments to the shape of the

histogram that you obtained in either part (b)(ii) or (b)(iii). [4]

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