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) Use it to fill the table below Step 1: Determine the class intervals and class widths from the histogram. The x-axis shows the class boundaries. The class width is the difference between the upper and lower class boundaries. Step 2: Read the frequency density for each class from the y-axis. Frequency density is given by the height of each bar. Step 3: Calculate the frequency for each class. Frequency = Frequency density × Class width. Step 4: Calculate the midpoint for each class. Midpoint = (Lower class boundary + Upper class boundary) / 2. | Classes | Midpoint (x) | Frequency Density | Class Width | Frequency (f) | | :-------- | :------------- | :---------------- | :---------- | :-------------- | | 19.5-29.5 | 24.5 | 0.4 | 10 | 0.4 × 10 = 4 | | 29.5-34.5 | 32.0 | 1.2 | 5 | 1.2 × 5 = 6 | | 34.5-54.5 | 44.5 | 1.1 | 20 | 1.1 × 20 = 22 | | 54.5-74.5 | 64.5 | 1.3 | 20 | 1.3 × 20 = 26 | | 74.5-84.5 | 79.5 | 0.6 | 10 | 0.6 × 10 = 6 | | 84.5-94.5 | 89.5 | 0.2 | 10 | 0.2 × 10 = 2 | The completed table is: |l|l|l| Classes & Midpoint & Frequency \\ 19.5-29.5 & 24.5 & 4 \\ 29.5-34.5 & 32.0 & 6 \\ 34.5-54.5 & 44.5 & 22 \\ 54.5-74.5 & 64.5 & 26 \\ 74.5-84.5 & 79.5 & 6 \\ 84.5-94.5 & 89.5 & 2 \\ b) Calculate the median Step 1: Calculate the total frequency (N) and the median position. Total frequency N = 4 + 6 + 22 + 26 + 6 + 2 = 66. Median position = (N)/(2) = (66)/(2) = 33^rd value. Step 2: Determine the median class. We need to find the class where the 33^rd value lies. Cumulative frequencies: 19.5-29.5: 4 29.5-34.5: 4+6=10 34.5-54.5: 10+22=32 54.5-74.5: 32+26=58 The 33^rd value falls into the class 54.5 - 74.5. This is the median class. Step 3: Apply the median formula for grouped data. The median formula is: Median = L + ((N)/(2) - CF_bf_m)c Where: L = Lower class boundary of the median class = 54.5 N = Total frequency = 66 CF_b = Cumulative frequency of the class before the median class = 32 f_m = Frequency of the median class = 26 c = Class width of the median class = 74.5 - 54.5 = 20 Substitute the values into the formula: Median = 54.5 + ((66)/(2) - 3226) × 20 Median = 54.5 + ((33 - 32)/(26)) × 20 Median = 54.5 + ((1)/(26)) × 20 Median = 54.5 + (20)/(26) Median = 54.5 + (10)/(13) Median ≈ 54.5 + 0.76923 Median ≈ 55.26923 Rounding to two decimal places: The median is 55.27. c) Calculate the mean Step 1: Calculate the sum of fx (frequency × midpoint) for all classes. | Classes | Midpoint (x) | Frequency (f) | fx | | :-------- | :------------- | :-------------- | :-------- | | 19.5-29.5 | 24.5 | 4 | 24.5 × 4 = 98.0 | | 29.5-34.5 | 32.0 | 6 | 32.0 × 6 = 192.0 | | 34.5-54.5 | 44.5 | 22 | 44.5 × 22 = 979.0 | | 54.5-74.5 | 64.5 | 26 | 64.5 × 26 = 1677.0 |