Step 1: Extract the data and calculate the necessary sums. The number of competitors is $n=10$. Let Dancing marks be $X$ and Fashion Parade marks be $Y$. | $X$ | $Y$ | $XY$ | $X^2$ | $Y^2$ | |-----|-----|------|-------|-------| | 10 | 18 | 180 | 100 | 324 | | 12 | 15 | 180 | 144 | 225 | | 16 | 10 | 160 | 256 | 100 | | 13 | 16 | 208 | 169 | 256 | | 17 | 19 | 323 | 289 | 361 | | 11 | 14 | 154 | 121 | 196 | | 18 | 13 | 234 | 324 | 169 | | 15 | 11 | 165 | 225 | 121 | | 19 | 17 | 323 | 361 | 289 | | 14 | 12 | 168 | 196 | 144 | |-----|-----|------|-------|-------| | $\sum X = 145$ | $\sum Y = 145$ | $\sum XY = 2075$ | $\sum X^2 = 2135$ | $\sum Y^2 = 2185$ | a) Calculate the product-moment correlation coefficient ($r$). The formula for the product-moment correlation coefficient is: $$r = \frac{n \sum XY - (\sum X)(\sum Y)}{\sqrt{[n \sum X^2 - (\sum X)^2][n \sum Y^2 - (\sum Y)^2]}}$$ Substitute the calculated sums into the formula: $$r = \frac{10(2075) - (145)(145)}{\sqrt{[10(2135) - (145)^2][10(2185) - (145)^2]}}$$ $$r = \frac{20750 - 21025}{\sqrt{[21350 - 21025][21850 - 21025]}}$$ $$r = \frac{-275}{\sqrt{[325][825]}}$$ $$r = \frac{-275}{\sqrt{268125}}$$ $$r = \frac{-275}{517.807879...}$$ $$r \approx -0.53108$$ Rounding to 2 decimal places: $$r \approx \boxed{-0.53}$$ b) Determine the least square regression line of the marks in fashion parade (Y) on the marks in dancing (X). The regression line is of the form $Y = a + bX$. The formula for the slope $b$ is: $$b = \frac{n \sum XY - (\sum X)(\sum Y)}{n \sum X^2 - (\sum X)^2}$$ Using the values calculated in part a): $$b = \frac{-275}{325}$$ $$b = -\frac{11}{13} \approx -0.8461538$$ The formula for the y-intercept $a$ is: $$a = \bar{Y} - b\bar{X}$$ First, calculate the means $\bar{X}$ and $\bar{Y}$: $$\bar{X} = \frac{\sum X}{n} = \frac{145}{10} = 14.5$$ $$\bar{Y} = \frac{\sum Y}{n} = \frac{145}{10} = 14.5$$ Now, calculate $a$: $$a = 14.5 - \left(-\frac{11}{13}\right)(14.5)$$ $$a = 14.5 + \frac{11 \times 14.5}{13}$$ $$a = 14.5 + \frac{159.5}{13}$$ $$a = 14.5 + 12