APPLICATION ASSIGNMENT 4 [TOTAL: 90 Marks] Question 1. The Leaf Data [37 Marks] A random sample of 25 leaves from an ivy plant is collected. For each leaf, the length (at its longest point) and the width (at its widest point) are measured. Length The data representing the width of leaves are shown in the 2nd column of Table 1; shaded green. The data representing the length of leaves are shown in the 3"d column of Table 1; shaded blue. Width For each leaf, the ratio of length to width is recorded in the 4th column of Table 1 and is shaded pink. Table 1 Table 2 Sample Width Length Length to Width Leaf Length (mm) Number mm) (mm) Ratio - Ranked in Order 1 52 28 0.54 2 31 17 0.55 3 56 35 0.63 4 67 45 0.67 5 63 48 0.76 6 40 33 0.83 17 56 47 0.84 8 57 52 0.91 9 39 39 1.00 10 51 51 1.00 11 55 56 1.02 12 36 38 1.06 13 44 47 1.07 14 42 46 1.10 15 70 78 1.11 16 61 69 1.13 17 50 60 1.20 18 55 67 1.22 19 52 66 1.27 20 67 87 1.30 21 60 78 22 66 89 1.35 23 38 53 1.39 24 80 115 1.44 25 75 110 1.47 L. The Leaf Length [7 Marks] (a) | In Table 2 above, using the column labelled 'Leaf Length (mm) - Ranked in Order', arrange the leaves in rank order of leaf length from shortest to longest. [1] (b) Use your rank order arrangement to find the median value of leaf length within the sample. [1] In the spare column of Table 2, draw a single arrow that points exactly to the median value.(c) Describe what this median value - of the length of leaves in the sample - represents. [1] (d) Calculate the mean value of the length of leaves in the sample, correct to one decimal place. [2] Describe what the mean value of the leaf length represents. (e) State the modal value/s of the leaf length (from Table 2) and explain what is meant by any [2] modal samples in this context. 2. The Leaf Ratio [7 Marks] (a) In Table 1, the column for Length to Width Ratio is missing one entry. Calculate the value of the missing entry. [1] (b) Regarding Length to Width Ratio, and considering the leaves with a ratio of less than 1, describe the relevant physical characteristics of these leaves. [1](c) Regarding Length to Width Ratio, and considering the leaves with a ratio that equals 1, [1] describe the relevant physical characteristics of these leaves. (d) Regarding Length to Width Ratio, and considering the leaves with a ratio of more than 1, [1] describe the relevant physical characteristics of these leaves. (e) Calculate the length to width ratio of a leaf of length 84 mm and width 21 mm. [3] Does this leaf "fit" with the sample of 25 leaves that you were given? Provide a practical, real-world, possible explanation behind the answer to this question. 3. The Report [10 Marks] As part of the report on the study of the relationship between length and width in the sample of 25 leaves, complete the following sentences, filling-in the blanks by referring back to Table 1 and Table 2. (a) [2] The range of lengths of the sample of 25 leaves is the difference between the longest leaf and the shortest leaf. The longest leaf is ....................... and the shortest leaf is ..................... The range is the difference between the two and this is .................. . (b [2] The average (mean) length of leaf is ..................... The middle (median) length is ................ The modal values of the sample of leaves are .........". ' (c [3] Approximately half (50%) of the sample is shorter than ...... . Approximately ............. % of the sample is longer than 78 mm. Approximately 67% of the sample leaves are shorter than .......... .(d) Determine the percentages to complete the sentence below, correct to one decimal place. [3] The percentage of the leaf sample that has a leaf ratio less than 1 is .................., while the percentage of the leaf sample which has leaf ratio greater than 1 is ................. . ............ % of the sample has leaf length equal to leaf width. 4. Plotting Points [7 Marks] (a) Plot the Samples Numbers 1, 10 and 24 (from Table 1) as points on the axes given below. Label each of the plotted points with its sample number. Let width, w, be the explanatory variable that is plotted on the X axis (the horizontal), and let length, L, be the response variable that is plotted on the Y axis (the vertical). [3] Scatterplot of Length of Ivy leaf versus its width. 120 100 80 Length of Leaf (mm) 60 20 20 40 60 80 100 120 W Width of Leaf (mm) The actual scatterplot, with all 25 of the Samples accurately plotted, is presented below. You should clearly see that the diagonal line drawn on the scatterplot is not the line of best fit for the sample points. (b) Use a ruler to draw in your line of best fit for all the data points, and then label it as "line of best fit". [2] Scatterplot of Length of Ivy leaf versus its width. 120 100 30 Length of Leaf (mm) 60 40 20 20 40 60 80 100 120 WFor all the data points, the association between length and width variables is evidently linear. [2] (c) Describe the strength and the direction of the association between leaf width and leaf length. 5. Prediction. [6 Marks] The linear equation that can be used for the purposes of predicting either the length (L) or the width (w) of any leaf of the ivy plant is: L = 1.58 w - 26 (a) Use the equation to predict the Length of a leaf if its width is 110 mm (to the nearest mm). [3] State whether this prediction is reasonable, and explain your reasoning. (b) Use the equation to predict the width of a leaf if its Length is 75 mm (to the nearest mm). [3] State whether this prediction reasonable, and explain your reasoning