This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Here are the steps to solve the problem: Step 1: Prepare the cumulative frequency table. To draw an ogive curve, we first need to calculate the upper class boundaries and the cumulative frequencies. | Height (cm) | Frequency (f) | Upper Class Boundary (UCB) | Cumulative Frequency (CF) | | :---------- | :------------ | :------------------------- | :------------------------ | | 130-139 | 10 | 139.5 | 10 | | 140-149 | 13 | 149.5 | 23 | | 150-159 | 16 | 159.5 | 39 | | 160-169 | 21 | 169.5 | 60 | | 170-179 | 18 | 179.5 | 78 | | 180-189 | 14 | 189.5 | 92 | | 190-199 | 8 | 199.5 | 100 | We also include a starting point for the ogive: (129.5, 0). Step 2: Draw the ogive curve (Part a). Draw a graph with "Height (cm)" on the x-axis (using the upper class boundaries) and "Cumulative Frequency" on the y-axis. The x-axis should range from approximately 129.5 to 199.5. The y-axis should range from 0 to 100. Plot the points: (129.5, 0), (139.5, 10), (149.5, 23), (159.5, 39), (169.5, 60), (179.5, 78), (189.5, 92), (199.5, 100). Connect these points with a smooth curve to form the ogive. Step 3: Estimate the median (Part b(i)). The median is the value at the 50th percentile. For a total of 100 students, the median position is (100)/(2) = 50. To find the median height, locate 50 on the cumulative frequency (y-axis), draw a horizontal line to the ogive, and then draw a vertical line down to the height (x-axis). Using interpolation from the table (as a proxy for reading from the graph): The 50th position falls in the 160-169 class. Median = L + ((N)/(2) - CF_bf_m) × c Where L = 159.5, N = 100, CF_b = 39, f_m = 21, c = 10. Median = 159.5 + ((50 - 39)/(21)) × 10 Median = 159.5 + ((11)/(21)) × 10 Median = 159.5 + 5.238 Median = 164.738 ≈ 164.74 cm Step 4: Determine the medium height if 30% of the candidates were below medium height (Part b(ii)). This asks for the height at the 30th percentile. The position is 30\% of 100 = 0.30 × 100 = 30. To find this height, locate 30 on the cumulative frequency (y-axis), draw a horizontal line to the ogive, and then draw a vertical line down to the height (x-axis). Using interpolation: The 30th position falls in the 150-159 class. Height = L + (Position - CF_bf_p) × c Where L = 149.5, Position = 30, CF_b = 23, f_p = 16, c = 10. Height = 149.5 + ((30 - 23)/(16)) × 10 Height = 149.5 + ((7)/(16)) × 10