1. **Problem Statement:** We have data on years of working experience and monthly salary for 7 employees. We need to: (i) Calculate the product moment correlation coefficient (Pearson's r) and interpret it. (ii) Find the least squares regression line of salary on years of experience. (iii) Estimate salary for 12 years experience and discuss accuracy. (iv) Calculate Spearman's rank correlation coefficient. Also, find the number of possible 4-member teams from 2 professors, 3 engineers, and 4 programmers if both professors must be selected. --- 2. **Given Data:** Years (X): 4, 1, 7, 3, 10, 6, 8 Salary (Y): 40, 25, 75, 32, 120, 75, 95 --- 3. **(i) Product Moment Correlation Coefficient (r):** Formula: $$r = \frac{n\sum XY - \sum X \sum Y}{\sqrt{(n\sum X^2 - (\sum X)^2)(n\sum Y^2 - (\sum Y)^2)}}$$ Calculate sums: $$\sum X = 4+1+7+3+10+6+8 = 39$$ $$\sum Y = 40+25+75+32+120+75+95 = 462$$ $$\sum XY = 4\times40 + 1\times25 + 7\times75 + 3\times32 + 10\times120 + 6\times75 + 8\times95 = 160 + 25 + 525 + 96 + 1200 + 450 + 760 = 3216$$ $$\sum X^2 = 4^2 + 1^2 + 7^2 + 3^2 + 10^2 + 6^2 + 8^2 = 16 + 1 + 49 + 9 + 100 + 36 + 64 = 275$$ $$\sum Y^2 = 40^2 + 25^2 + 75^2 + 32^2 + 120^2 + 75^2 + 95^2 = 1600 + 625 + 5625 + 1024 + 14400 + 5625 + 9025 = 37824$$ Calculate numerator: $$7 \times 3216 - 39 \times 462 = 22512 - 18018 = 4494$$ Calculate denominator: $$\sqrt{(7 \times 275 - 39^2)(7 \times 37824 - 462^2)} = \sqrt{(1925 - 1521)(264768 - 213444)} = \sqrt{404 \times 51324}$$ $$= \sqrt{20734496} \approx 4553.8$$ Therefore, $$r = \frac{4494}{4553.8} \approx 0.987$$ **Interpretation:** The correlation coefficient is approximately 0.987, indicating a very strong positive linear relationship between years of experience and monthly salary. --- 4. **(ii) Least Squares Regression Line:** Formula: $$y = a + bx$$ where $$b = \frac{n\sum XY - \sum X \sum Y}{n\sum X^2 - (\sum X)^2}$$ $$a = \bar{y} - b\bar{x}$$ Calculate slope $b$: $$b = \frac{7 \times 3216 - 39 \times 462}{7 \times 275 - 39^2} = \frac{4494}{404} \approx 11.12$$ Calculate means: $$\bar{x} = \frac{39}{7} \approx 5.57$$ $$\bar{y} = \frac{462}{7} = 66$$ Calculate intercept $a$: $$a = 66 - 11.12 \times 5.57 = 66 - 61.91 = 4.09$$ Regression line: $$y = 4.09 + 11.12x$$ --- 5. **(iii) Estimate Salary for 12 Years Experience:** Substitute $x=12$: $$y = 4.09 + 11.12 \times 12 = 4.09 + 133.44 = 137.53$$ **Accuracy and Reliability:** Since 12 years is slightly outside the observed range (max 10), prediction is an extrapolation and may be less reliable. However, given the strong correlation, the estimate is reasonable but should be treated cautiously. --- 6. **(iv) Spearman's Rank Correlation Coefficient:** Rank the data: Years (X): 1(1), 3(2), 4(3), 6(4), 7(5), 8(6), 10(7) Salary (Y): 25(1), 32(2), 40(3), 75(4.5), 75(4.5), 95(6), 120(7) Calculate difference in ranks $d_i$ and $d_i^2$: |X rank|Y rank|$d_i$|$d_i^2$| |---|---|---|---| |3|3|0|0| |1|1|0|0| |5|4.5|0.5|0.25| |2|2|0|0| |7|7|0|0| |4|4.5|-0.5|0.25| |6|6|0|0| Sum of $d_i^2 = 0.5$ Formula: $$r_s = 1 - \frac{6 \sum d_i^2}{n(n^2 - 1)} = 1 - \frac{6 \times 0.5}{7(49 - 1)} = 1 - \frac{3}{7 \times 48} = 1 - \frac{3}{336} = 1 - 0.0089 = 0.991$$ Spearman's rank correlation coefficient is approximately 0.991, indicating a very strong monotonic relationship. --- 7. **Number of Possible Teams:** We must select 4 members including both professors (2 professors). Remaining members to select: 4 - 2 = 2 Available non-professors: 3 engineers + 4 programmers = 7 Number of ways to select 2 from 7: $$\binom{7}{2} = \frac{7 \times 6}{2 \times 1} = 21$$ **Final answers:** - Product moment correlation coefficient $r \approx 0.987$ - Regression line $y = 4.09 + 11.12x$ - Estimated salary for 12 years $\approx 137.53$ - Spearman's rank correlation coefficient $r_s \approx 0.991$ - Number of possible teams = 21