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.)

Solution Steps

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

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

For the given data, the calculated sample covariance is:

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

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

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

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

The calculated standard deviations are:

$$\sigma_X = 5.429$$

$$\sigma_Y = 10.858$$

The correlation coefficient $r$ is calculated using the formula:

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

Substituting the values:

$$r = \frac{58.945}{5.429 \times 10.858} = 1.0$$

Final Answer