Chapter 1 Exercise 3 & 4 Notes

Image Analysis Visual Elements: Text: The majority of the image is filled with text, including headings, mathematical equations, and explanations. Table: A table presents the sample cross-classified according to the Working Environment Index and Attitude toward early retirement. Bar Chart: A bar chart visually represents the conditional distributions of the Attitude toward early retirement. Text Transcription: Exercise 3. In a 2002 study of attitudes toward early retirement and partial pension, a special index for working environment is reported for each interviewed person. The index takes values from 0 to 10, with 0 representing an excellent working environment and 10 a very bad working environment. The table below shows the sample cross-classified according to this index and according to attitude toward early retirement.   Attitude toward early retirement 0-2 3-6 7-10 Total Good system 267 152 52 471 Maybe good system 30 11 24 65 Bad system 20 24 10 54 Total 317 187 86 590 a. Construct the marginal frequency distribution of the Working Environment Index. b. What is the frequency of being in the class [7,10] of the index and considering the early retirement a good system? What is the frequency of being in the class [3,6] and considering the early retirement a bad or maybe a good system? c. Build up a cluster bar chart and a component bar chart for the conditional distributions of the Attitude toward early retirement. d. Does the attitude toward early retirement have a connection with the working environment? a. The frequency distribution of one categorical variable regardless of the other one is a marginal frequency distribution. In this case we consider the column sum (267+68+26)+(152+30+24)+(52+11+40)=590 b. The frequency of being in [7,10] and considering the early retirement a good system is 52. The frequency of being in [3,6] and considering the early retirement a bad or maybe a good system is 30+24=54 ("or" implies summation!) c. We have to compute conditional relative frequencies of Attitude toward early retirement conditional to each of the three modalities of the Working Environment Index. We have: Attitude toward early retirement 0-2 3-6 7-10 Good system 0.74 0.74 0.71 Maybe good system 0.19 0.14 0.45 Bad system 0.07 0.12 0.14 Total 1 1 1 Exercise 4. In the following table are reported the marks of (n=8) students to the exams of mathematics and statistics. Student Math (X) Stats (Y) (x - \overline{x}) (y - \overline{y}) ((x - \overline{x})^2) ((y - \overline{y})^2) ((x - \overline{x})(y - \overline{y})) 1 24 23 -0.63 -2.38 0.3969 5.6644 1.5034 2 27 28 2.37 2.62 5.6169 6.8600 6.1094 3 30 30 5.37 4.62 28.8369 21.3449 24.8769 4 26 27 1.37 1.62 1.8769 2.6244 2.2144 5 29 30 4.37 4.62 19.0969 21.3449 20.1894 6 18 20 -6.63 -5.38 43.9569 28.9444 35.6694 7 21 20 -3.63 -5.38 13.1769 28.9444 19.5294 8 22 25 -1.63 -0.38 2.6569 0.1444 0.6206 Compute a measure to evaluate the linear relationship between the two marks. [r(x,y) = \frac{cov(x,y)}{S_x S_y}] [\overline{x} = \sum_{i=1}^{n} \frac{x_i}{N} = \frac{24+27+30+26+29+18+21+22}{8} = 24.63] [\overline{y} = \sum_{i=1}^{n} \frac{y_i}{N} = \frac{23+28+30+27+30+20+20+25}{8} = 25.38] [s_x = \sqrt{\frac{\sum_{i=1}^{n} (x_i - \overline{x})^2}{N}} = \sqrt{\frac{115.8352}{8}} = 3.87] [s_y = \sqrt{\frac{\sum_{i=1}^{n} (y_i - \overline{y})^2}{N}} = \sqrt{\frac{135.2}{8}} = 4.11] [cov(x,y) = \frac{\sum_{i=1}^{n} (x_i - \overline{x})(y_i - \overline{y})}{N} = \frac{111.1252}{8} = 13.89] Correlation coefficient [r = \frac{13.89}{(3.87)(4.11)} = 0.94] Since r varies between -1 and 1, this result suggests a nearly perfect linear relationship between the marks in mathematics and the marks in statistics.