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
a) Construct a cumulative frequency table for the distribution. Step 1: Determine the upper class boundaries for each weight interval. The class intervals are given as 40-44, 45-49, etc. To make them continuous for a cumulative frequency curve, we use upper class boundaries. For example, for 40-44, the upper boundary is 44.5 (midpoint between 44 and 45). Step 2: Calculate the cumulative frequency by adding the frequencies sequentially. The cumulative frequency table is as follows: | Weight (kg) | Upper Class Boundary (kg) | Frequency | Cumulative Frequency | |-------------|---------------------------|-----------|----------------------| | 40-44 | 44.5 | 5 | 5 | | 45-49 | 49.5 | 6 | 11 | | 50-54 | 54.5 | 10 | 21 | | 55-59 | 59.5 | 4 | 25 | | 60-64 | 64.5 | 5 | 30 | | 65-69 | 69.5 | 3 | 33 | | 70-74 | 74.5 | 2 | 35 | b) Draw a cumulative frequency curve for the distribution. Step 1: Plot the points using the upper class boundaries on the x-axis and the cumulative frequencies on the y-axis. The points to plot are: (44.5, 5) (49.5, 11) (54.5, 21) (59.5, 25) (64.5, 30) (69.5, 33) (74.5, 35) It is also good practice to start the curve from the lower boundary of the first class with a cumulative frequency of 0. The lower boundary for 40-44 is 39.5, so plot (39.5, 0). Step 2: Draw a smooth curve connecting these points. (As an AI, I cannot draw the curve, but the description above outlines how to construct it.) c) Use your curve to estimate the median. Step 1: Determine the position of the median. The total number of students (N) is 35. The median is the value at the (N)/(2)-th position. Median position = (35)/(2) = 17.5 Step 2: Locate the median position on the cumulative frequency axis (y-axis) of your curve. Draw a horizontal line from 17.5 on the y-axis to intersect the cumulative frequency curve. Step 3: Draw a vertical line from the intersection point down to the weight axis (x-axis). Read the value on the x-axis. This value is the estimated median. Based on the cumulative frequency table, the 17.5th value falls within the 50-54 kg class (where cumulative frequency goes from 11 to 21). Using interpolation for a precise estimate (which would be read from a well-drawn curve): Lower boundary of median class (L) = 49.5 Cumulative frequency before median class (CF) = 11 Frequency of median class (f) = 10 Class width (c) = 5 Median = L + ((N)/(2) - CFf) × c Median = 49.5 + ((17.5 - 11)/(10)) × 5 Median = 49.5 + ((6.5)/(10)) × 5 Median = 49.5 + 0.65 × 5 Median = 49.5 + 3.25 Median = 52.75 kg (Your graphical estimate from the curve should be close to this value.) What's next?