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
Okay, Trinity🦋💓, let's break down this statistics problem. Part 1: Calculate the standard deviation This calculation uses the first table provided in the image: | Number of fruits per tree (x) | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | |---------------------------------|---|---|---|---|---|---|---|---|---| | Frequency (f) | 1 | 3 | 4 | 6 | 10 | 16 | 12 | 8 | 0 | Step 1: Calculate the total frequency ( f), sum of xf ( xf), and sum of x^2f ( x^2f). f = 1+3+4+6+10+16+12+8+0 = 60 xf = (0 × 1) + (1 × 3) + (2 × 4) + (3 × 6) + (4 × 10) + (5 × 16) + (6 × 12) + (7 × 8) + (8 × 0) xf = 0 + 3 + 8 + 18 + 40 + 80 + 72 + 56 + 0 = 277 x^2f = (0^2 × 1) + (1^2 × 3) + (2^2 × 4) + (3^2 × 6) + (4^2 × 10) + (5^2 × 16) + (6^2 × 12) + (7^2 × 8) + (8^2 × 0) x^2f = 0 + 3 + 16 + 54 + 160 + 400 + 432 + 392 + 0 = 1457 Step 2: Calculate the mean (x). x = ( xf)/( f) = (277)/(60) ≈ 4.6167 Step 3: Calculate the standard deviation (). The formula for standard deviation is = sqrt(( x^2f)/( f) - (( xf)/( f))^2). = sqrt((1457)/(60) - ((277)/(60))^2) = sqrt(24.283333 - (4.616667)^2) = sqrt(24.283333 - 21.313611) = sqrt(2.969722) ≈ 1.723 The standard deviation is 1.723. Part 2: Complete the table and draw the graph There is an inconsistency between the first table and the values given in the second table. We will assume the given Cumulative Frequency (CF) and Relative Cumulative Frequency (RCF) for x=3 in the second table are correct to determine the total frequency, and then complete the table consistently. Step 1: Determine the total frequency. From the row for x=3 in the second table, CF = 10 and RCF = 0.10. Since RCF = CF / Total Frequency, we have: Total Frequency = CFRCF = (10)/(0.10) = 100 Step 2: Complete the table using Total Frequency = 100. • For x=2: RCF = 0.02. CF (c) = RCF × Total Frequency = 0.02 × 100 = 2. • For x=4: CF = 16. RCF = CF / Total Frequency = 16 / 100 = 0.16. (Note: The question's table shows 0.17, but 0.16 is consistent with a total frequency of 100). • For x=5: CF = 42. RCF = CF / Total Frequency = 42 / 100 = 0.42. • For x=7: CF = 60. RCF = CF / Total Frequency = 60 / 100 = 0.60. • For x=8: CF = 60. RCF = CF / Total Frequency = 60 / 100 = 0.60. The completed table is: | Number of fruits per tree (x) | 2 | 3 | 4 | 5 | 7 | 8 | |---------------------------------|---|---|---|---|---|---| | Cumulative Frequency (CF) | 2 | 10 | 16 | 42 | 60 | 60 | | Relative cumulative frequency | 0.02 | 0.10 | 0.16 | 0.42 | 0.60 | 0.60 | Step 3: Instructions for drawing the relative cumulative frequency curve. Plot the points (upper class boundary, relative cumulative frequency). For discrete data, the upper class boundary for an x value is x+0.5. The points to plot are: • (1.5, 0) (starting point, assuming no fruits below x=2) • (2.5, 0.02) • (3.5, 0.10) • (4.5, 0.16) • (5.5, 0.42) • (7.5, 0.60) • (8.5, 0.60) Use the given scales: • X-axis: 2 cm to represent 1 unit for 0 x 8. • Y-axis: 2 cm to represent 0.1 for relative cumulative frequency. Draw a smooth curve connecting these points. Part 3: Use the graph to find the 80th percentile Step 1: Locate the 80th percentile on the y-axis. The 80th percentile corresponds to a relative cumulative frequency of 0.80. Step 2: Read the value from the graph. On your drawn graph, find 0.80 on the y-axis. Draw a horizontal line from y=0.80 until it intersects the relative cumulative frequency curve. From the intersection point, draw a vertical line down to the x-axis. The value where this vertical line intersects the x-axis is the 80th percentile. Since I cannot draw the graph, I cannot provide the numerical answer for the 80th percentile. 3 done, 2 left today. You're making progress.