University:
York University
Course:
MATH 1131 | Introduction to Statistics I
Academic year:

2022
Views:

380

Pages:

2
Author:

Niklas

Exercise 9: Analysis of Company Sizes Problem: The table shows the densities of relative frequencies of 216 companies in Friuli-Venezia Giulia, classified according to the number of employees. Tasks: Complete the table. Display data graphically. Plot the cumulative line-graph for these data. Given: Number of Employees (N) Density (d_i) (3-10) 0.112024 (10-20) ? (20-50) 0.001767 (50-100) 0.000285 Solution: Part a: Completing the Table We know that w_2 = 10. To find f_2, we use the inverse formula of the density: f_i = d_i * w_i We also know that the sum of the relative frequencies is 1, so we subtract f_1, f_3, and f_4 from 1 to find f_2. f_1 = d_1 * w_1 = 0.112024 * 7 = 0.788788 f_3 = d_3 * w_3 = 0.001767 * 30 = 0.05301 f_4 = d_4 * w_4 = 0.000285 * 50 = 0.01425 f_2 = 1 - f_1 - f_3 - f_4 = 1 - 0.788788 - 0.05301 - 0.01425 = 0.143952 Now we can find d_2: d_2 = f_2 / w_2 = 0.143952 / 10 = 0.0143952 Part b: Displaying Data Graphically To display the data graphically, we use a histogram and a cumulative line-graph. Cumulative Line-Graph: Number of Employees (N) f_i F(x) (3-10) 0.788788 0.788788 (10-20) 0.143952 0.93274 (20-50) 0.05301 0.98575 (50-100) 0.01425 1 The cumulative distribution function (CDF) F(x) is defined as follows: F(x) = 0 if x < 3 F(x) = 0.788788 if 3 <= x < 10 F(x) = 0.93274 if 10 <= x < 20 F(x) = 0.98575 if 20 <= x < 50 F(x) = 1 if 50 <= x The histogram and cumulative line-graph would be plotted based on these values. Exercise 10: Obtaining Frequency Distribution from ECDF Problem: Obtain the frequency distribution from the empirical cumulative distribution function (ECDF) shown in the figure. Given: N = 10 F(x) = 0 if x < 1 F(x) = 0.179 if 1 <= x < 2 F(x) = 0.339 if 2 <= x < 3 F(x) = 0.482 if 3 <= x < 4 F(x) = 0.643 if 4 <= x < 5 F(x) = 0.786 if 5 <= x < 6 F(x) = 0.911 if 6 <= x < 7 F(x) = 1 if 7 <= x < 8 Solution: Step 1: Understanding the ECDF The ECDF represents the cumulative probability of a variable being less than or equal to a certain value. Step 2: Calculating Absolute Frequencies We can obtain the absolute frequencies using the following formula: n_i = N * (F(x_i) - F(x_{i-1})) where: n_i is the absolute frequency of the i-th class N is the total number of observations F(x_i) is the ECDF value at the upper bound of the i-th class F(x_{i-1}) is the ECDF value at the lower bound of the i-th class Step 3: Calculating Absolute Frequencies for Each Class Class 1: n_1 = 10 * (0.179 - 0) = 1.79 Class 2: n_2 = 10 * (0.339 - 0.179) = 1.6 Similarly, we can calculate the absolute frequencies for the remaining classes.