1. **Problem Statement:** We have marks scored by 40 pupils in English and Mathematics tests grouped into intervals. We need to: (a) Draw two histograms on the same axes for English and Mathematics marks. (b) Draw two frequency polygons on the same axes for the same data. (c) Calculate the mean mark for each subject. (d) Compare the scores in Mathematics and English. 2. **Data given:** Marks intervals: 0-9, 10-19, 20-29, 30-39, 40-49, 50-59, 60-69, 70-79, 80-89, 90-99 Frequencies for English: 2, 6, 6, 4, 6, 11, 2, 2, 1, 0 Frequencies for Mathematics: 3, 3, 2, 5, 7, 12, 3, 2, 2, 1 3. **Step (a) and (b): Drawing histograms and frequency polygons** - Histograms: Bars for each interval with heights equal to frequencies for English and Mathematics. - Frequency polygons: Plot points at midpoints of intervals with frequencies and connect with lines. Midpoints of intervals are calculated as: $$\text{Midpoint} = \frac{\text{Lower limit} + \text{Upper limit}}{2}$$ Midpoints: $$4.5, 14.5, 24.5, 34.5, 44.5, 54.5, 64.5, 74.5, 84.5, 94.5$$ 4. **Step (c): Calculate mean marks** Mean formula for grouped data: $$\bar{x} = \frac{\sum f_i x_i}{\sum f_i}$$ where $f_i$ is frequency and $x_i$ is midpoint. Calculate for English: $$\sum f_i x_i = 2\times4.5 + 6\times14.5 + 6\times24.5 + 4\times34.5 + 6\times44.5 + 11\times54.5 + 2\times64.5 + 2\times74.5 + 1\times84.5 + 0\times94.5$$ $$= 9 + 87 + 147 + 138 + 267 + 599.5 + 129 + 149 + 84.5 + 0 = 1609$$ Total frequency = 40 Mean English: $$\bar{x}_{English} = \frac{1609}{40} = 40.225$$ Calculate for Mathematics: $$\sum f_i x_i = 3\times4.5 + 3\times14.5 + 2\times24.5 + 5\times34.5 + 7\times44.5 + 12\times54.5 + 3\times64.5 + 2\times74.5 + 2\times84.5 + 1\times94.5$$ $$= 13.5 + 43.5 + 49 + 172.5 + 311.5 + 654 + 193.5 + 149 + 169 + 94.5 = 1849$$ Mean Mathematics: $$\bar{x}_{Math} = \frac{1849}{40} = 46.225$$ 5. **Step (d): Interpretation** - The mean mark in Mathematics (46.225) is higher than in English (40.225). - Mathematics scores are generally higher, indicating better performance or easier test. - Histograms and frequency polygons would show distribution shapes and frequency differences visually.