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 calculations for the mean, median, and mode: Step 1: Create an extended frequency distribution table. First, we need to determine the midpoints (x) for each class interval, calculate Fx, and find the cumulative frequencies (Cf). The class width (w) is 10 (e.g., 9-0+1=10). The lower class boundaries (L) for calculations are found by subtracting 0.5 from the lower limit of each class (e.g., 20-0.5=19.5). | Age of building | F (Frequency) | Midpoint (x) | Fx | Cf (Cumulative Frequency) | |:---------------:|:-------------:|:------------:|:------:|:-------------------------:| | 0-9 | 19 | 4.5 | 85.5 | 19 | | 10-19 | 47 | 14.5 | 681.5 | 66 | | 20-29 | 105 | 24.5 | 2572.5 | 171 | | 30-39 | 93 | 34.5 | 3208.5 | 264 | | 40-49 | 16 | 44.5 | 712.0 | 280 | | Total | N = 280 | | ΣFx = 7260 | | 1. Calculate the Mean Step 2: Use the formula for the mean of grouped data. The mean (x) is given by: x = ( Fx)/(N) Substitute the total sum of Fx and the total frequency (N) from the table: x = (7260)/(280) x ≈ 25.93 The mean age of the hospital buildings is 25.93 years. 2. Calculate the Median Step 3: Determine the median class and apply the median formula. The position of the median is (N)/(2) = (280)/(2) = 140. The median class is the class where the cumulative frequency (Cf) first exceeds 140. From the table, this is the 20-29 class (Cf = 171). For the median class (20-29): • Lower class boundary (L) = 20 - 0.5 = 19.5 • Cumulative frequency of the class before the median class (Cf_b) = 66 • Frequency of the median class (f_m) = 105 • Class width (w) = 10 The formula for the median of grouped data is: Median = L + ((N)/(2) - Cf_bf_m) × w Substitute the values: Median = 19.5 + ((140 - 66)/(105)) × 10 Median = 19.5 + ((74)/(105)) × 10 Median = 19.5 + 0.70476 × 10 Median = 19.5 + 7.0476 Median ≈ 26.55 The median age of the hospital buildings is 26.55 years. 3. Calculate the Mode Step 4: Determine the modal class and apply the mode formula. The modal class is the class with the highest frequency. From the table, the highest frequency is 105, which corresponds to the 20-29 class. For the modal class (20-29): • Lower class boundary (L) = 20 - 0.5 = 19.5 • Frequency of the modal class (f_m) = 105 • Frequency of the class before the modal class (f_1) = 47 (for 10-19) • Frequency of the class after the modal class (f_2) = 93 (for 30-39) • Class width (w) = 10 The formula for the mode of grouped data is: Mode = L + ((f_m - f_1)/((f_m - f_1) + (f_m - f_2))) × w Substitute the values: Mode = 19.5 + ((105 - 47)/((105 - 47) + (105 - 93))) × 10 Mode = 19.5 + ((58)/(58 + 12)) × 10 Mode = 19.5 + ((58)/(70)) × 10 Mode = 19.5 + 0.82857 × 10 Mode = 19.5 + 8.2857 Mode ≈ 27.79 The mode age of the hospital buildings is 27.79 years. That's 2 down. 3 left today — send the next one.