To find the Covariance and Coefficient of Correlation between $X$ and $Y$, let's proceed step by step:

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Given Data:

$X = [15, 29, 87, 11, 43]$ $Y = [34, 10, 30, 22, 50]$

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Step 1: Compute Mean of X and Y

The formula for the mean: $\bar{X} = \frac{\sum X_i}{n}, \quad \bar{Y} = \frac{\sum Y_i}{n}$ where $n$ is the number of data points (5 in this case).

$\bar{X} = \frac{15 + 29 + 87 + 11 + 43}{5} = \frac{185}{5} = 37$ $\bar{Y} = \frac{34 + 10 + 30 + 22 + 50}{5} = \frac{146}{5} = 29.2$

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Step 2: Compute Deviations

Calculate the deviations from the mean for both $X$ and $Y$: $dX_i = X_i - \bar{X}, \quad dY_i = Y_i - \bar{Y}$

$X_i$	$Y_i$	$dX_i = X_i - \bar{X}$	$dY_i = Y_i - \bar{Y}$	$dX_i \cdot dY_i$	$dX_i^2$	$dY_i^2$
15	34	$15 - 37 = -22$	$34 - 29.2 = 4.8$	$-22 \cdot 4.8 = -105.6$	484	23.04
29	10	$29 - 37 = -8$	$10 - 29.2 = -19.2$	$-8 \cdot -19.2 = 153.6$	64	368.64
87	30	$87 - 37 = 50$	$30 - 29.2 = 0.8$	$50 \cdot 0.8 = 40$	2500	0.64
11	22	$11 - 37 = -26$	$22 - 29.2 = -7.2$	$-26 \cdot -7.2 = 187.2$	676	51.84
43	50	$43 - 37 = 6$	$50 - 29.2 = 20.8$	$6 \cdot 20.8 = 124.8$	36	432.64

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Step 3: Compute Covariance

The formula for covariance is: $\text{Cov}(X, Y) = \frac{\sum (dX_i \cdot dY_i)}{n}$

$\sum (dX_i \cdot dY_i) = -105.6 + 153.6 + 40 + 187.2 + 124.8 = 400$

$\text{Cov}(X, Y) = \frac{400}{5} = 80$

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Step 4: Compute Coefficient of Correlation (r)

The formula for the correlation coefficient is: $r = \frac{\text{Cov}(X, Y)}{\sqrt{\text{Var}(X) \cdot \text{Var}(Y)}}$

$\text{Var}(X) = \frac{\sum dX_i^2}{n}, \quad \text{Var}(Y) = \frac{\sum dY_i^2}{n}$

$\sum dX_i^2 = 484 + 64 + 2500 + 676 + 36 = 3760, \quad \text{Var}(X) = \frac{3760}{5} = 752$ $\sum dY_i^2 = 23.04 + 368.64 + 0.64 + 51.84 + 432.64 = 876.8, \quad \text{Var}(Y) = \frac{876.8}{5} = 175.36$

$r = \frac{80}{\sqrt{752 \cdot 175.36}} = \frac{80}{\sqrt{131936.32}} = \frac{80}{363.2} \approx 0.22$

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Final Results:

1. Covariance: $\text{Cov}(X, Y) = 80$
2. Correlation Coefficient: $r \approx 0.22$

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Do you want me to explain any specific part or further verify these calculations?

Related Questions:

1. How is the correlation coefficient affected by outliers in data?
2. Can covariance be negative, and what does it imply in practical terms?
3. What is the difference between covariance and variance?
4. How would adding a constant to all $X$ values affect covariance and correlation?
5. What are the limitations of using the correlation coefficient?

Tip: The correlation coefficient $r$ ranges between $-1$ and $+1$. An $r$ value close to 0 indicates little to no linear relationship between the variables.