Questions: The following is a set of data from a sample of n=11 items. Complete parts (a) through (c). X 20 6 11 14 13 17 16 12 10 1 7 Y 40 12 22 28 26 34 32 24 20 2 14 a. Compute the sample covariance. (Round to three decimal places as needed.)

The following is a set of data from a sample of n=11 items. Complete parts (a) through (c).

X 20 6 11 14 13 17 16 12 10 1 7
Y 40 12 22 28 26 34 32 24 20 2 14

a. Compute the sample covariance.
(Round to three decimal places as needed.)
Transcript text: The following is a set of data from a sample of $n=11$ items. Complete parts (a) through (c). \[ \begin{array}{rrrrrrrrrrrr} \mathbf{X} & 20 & 6 & 11 & 14 & 13 & 17 & 16 & 12 & 10 & 1 & 7 \\ \mathbf{Y} & 40 & 12 & 22 & 28 & 26 & 34 & 32 & 24 & 20 & 2 & 14 \end{array} \] a. Compute the sample covariance. $\square$ (Round to three decimal places as needed.)
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Solution

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Solution Steps

Step 1: Calculate the Sample Covariance

The sample covariance between the variables X X and Y Y is calculated using the formula:

Cov(X,Y)=1n1i=1n(XiXˉ)(YiYˉ) \text{Cov}(X,Y) = \frac{1}{n-1} \sum_{i=1}^{n} (X_i - \bar{X})(Y_i - \bar{Y})

For the given data, the calculated sample covariance is:

Cov(X,Y)=58.945 \text{Cov}(X,Y) = 58.945

Step 2: Calculate the Standard Deviations

The standard deviation for X X and Y Y is calculated using the formulas:

σX=1n1i=1n(XiXˉ)2 \sigma_X = \sqrt{\frac{1}{n-1} \sum_{i=1}^{n} (X_i - \bar{X})^2}

σY=1n1i=1n(YiYˉ)2 \sigma_Y = \sqrt{\frac{1}{n-1} \sum_{i=1}^{n} (Y_i - \bar{Y})^2}

The calculated standard deviations are:

σX=5.429 \sigma_X = 5.429

σY=10.858 \sigma_Y = 10.858

Step 3: Calculate the Correlation Coefficient

The correlation coefficient r r is calculated using the formula:

r=Cov(X,Y)σXσY r = \frac{\text{Cov}(X,Y)}{\sigma_X \sigma_Y}

Substituting the values:

r=58.9455.429×10.858=1.0 r = \frac{58.945}{5.429 \times 10.858} = 1.0

Final Answer

The sample covariance is 58.945 \boxed{58.945} and the correlation coefficient is 1.0 \boxed{1.0} .

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