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.
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Question 4.1.1: Determine the missing values in the table above. Step 1: Calculate missing value A (Frequency). For the first row, "MIDPOINT X FREQUENCY" is 20 and "MIDPOINT" is 10. A × 10 = 20 A = (20)/(10) A = 2 Step 2: Calculate missing value B (Midpoint). For the third row, the rainfall interval is 40 < x 60. The midpoint is the average of the lower and upper bounds. B = (40 + 60)/(2) B = (100)/(2) B = 50 Question 4.1.2: Using the table above, calculate the estimated mean rainfall. Step 1: Complete the sums for frequency and (midpoint × frequency). Using A=2 and B=50: Total Frequency ( f): 2 + 4 + 5 + 6 + 8 + 3 + 2 = 30 Total (Midpoint × Frequency) ( (x · f)): 20 + 120 + (50 × 5) + 420 + 720 + 330 + 260 20 + 120 + 250 + 420 + 720 + 330 + 260 = 2120 Step 2: Calculate the estimated mean rainfall (x). The formula for the estimated mean is: x = ( (x · f))/( f) x = (2120)/(30) x = 70.67 mm Question 4.1.3: Draw an Ogive (cumulative frequency graph) of the data on the GRID PROVIDED. I cannot draw a graph directly. You would plot the cumulative frequency against the upper class boundaries. Question 4.1.4: Write the following: a) Modal class Step 1: Identify the highest frequency. The frequencies are 2, 4, 5, 6, 8, 3, 2. The highest frequency is 8. Step 2: Determine the class interval corresponding to the highest frequency. The class interval with a frequency of 8 is 80 < x 100. The modal class is 80 < x 100. b) Estimated median rainfall Step 1: Determine the position of the median. Total frequency (N) = 30. The median position is (N)/(2) = (30)/(2) = 15. The median is the 15th value. Step 2: Identify the median class. Calculate cumulative frequencies: 0 < x 20: Cum Freq = 2 20 < x 40: Cum Freq = 2 + 4 = 6 40 < x 60: Cum Freq = 6 + 5 = 11 60 < x 80: Cum Freq = 11 + 6 = 17 The 15th value falls into the class 60 < x 80, as its cumulative frequency (17) is the first to exceed the median position (15). So, the median class is 60 < x 80. Step 3: Calculate the estimated median using the formula for grouped data. Median = L + ( (N)/(2) - CF_bf_m ) × w Where: L = 60 (lower boundary of the median class) N = 30 (total frequency) CF_b = 11 (cumulative frequency of the class before the median class) f_m = 6 (frequency of the median class) w = 20 (class width, 80 - 60) Median = 60 + ( (30)/(2) - 116 ) × 20 Median = 60 + ( (15 - 11)/(6) ) × 20 Median = 60 + ( (4)/(6) ) × 20 Median = 60 + ( (2)/(3) ) × 20 Median = 60 + (40)/(3) Median = 60 + 13.333... Median = 73.33 mm 3 done, 2 left today. You're making progress.