Calculate the product-moment correlation coefficient r and the least squares regression line of Y on X.

a) To calculate the product-moment correlation coefficient $r$, we first need to find the sums: $\sum X$, $\sum Y$, $\sum X^2$, $\sum Y^2$, and $\sum XY$.
There are $n=10$ competitors.

Step 1: Calculate the sums.

$$\sum X = 10+12+16+13+17+11+18+15+19+14 = 145$$ $$\sum Y = 18+15+10+16+19+14+13+11+17+12 = 145$$ $$\sum X^2 = 10^2+12^2+16^2+13^2+17^2+11^2+18^2+15^2+19^2+14^2 = 2185$$ $$\sum Y^2 = 18^2+15^2+10^2+16^2+19^2+14^2+13^2+11^2+17^2+12^2 = 2185$$ $$\sum XY = (10 \times 18) + (12 \times 15) + (16 \times 10) + (13 \times 16) + (17 \times 19) + (11 \times 14) + (18 \times 13) + (15 \times 11) + (19 \times 17) + (14 \times 12) = 2095$$

Step 2: Apply the formula for the product-moment correlation coefficient.

$$r = \frac{n \sum XY - (\sum X)(\sum Y)}{\sqrt{[n \sum X^2 - (\sum X)^2][n \sum Y^2 - (\sum Y)^2]}}$$

Substitute the calculated values:

$$r = \frac{10(2095) - (145)(145)}{\sqrt{[10(2185) - (145)^2][10(2185) - (145)^2]}}$$ $$r = \frac{20950 - 21025}{\sqrt{[21850 - 21025][21850 - 21025]}}$$ $$r = \frac{-75}{\sqrt{[825][825]}}$$ $$r = \frac{-75}{825}$$ $$r = -0.090909...$$

Rounding to 2 decimal places:

$$r = -0.09$$

The product-moment correlation coefficient is $\boxed{-0.09}$.

b) To determine the least square regression line of the marks in fashion parade (Y) on the marks in dancing (X), we use the equation $Y = a + bX$.

Step 1: Calculate the slope $b$.

$$b = \frac{n \sum XY - (\sum X)(\sum Y)}{n \sum X^2 - (\sum X)^2}$$

Using the values from part a):

$$b = \frac{-75}{825}$$ $$b = -\frac{1}{11}$$

Step 2: Calculate the y-intercept $a$.
First, find 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, use the formula $a = \bar{Y} - b\bar{X}$:

$$a = 14.5 - \left(-\frac{1}{11}\right)(14.5)$$ $$$$