The problem is asking to find the covariance of the given data set of paired values:

$(1, 5), (2, 7), (3, 9), (4, 11), (5, 10), (6, 9), (7, 8), (8, 7), (9, 6), (10, 5)$

Covariance Formula:

To calculate the covariance between two sets of data $X$ and $Y$, we use the following formula:

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

Where:

• $X_i$ and $Y_i$ are the individual data points,
• $\bar{X}$ and $\bar{Y}$ are the means of the data sets $X$ and $Y$,
• $n$ is the number of data points.

Step-by-Step Solution:

Step 1: Extract the data

The data consists of paired values $(X_i, Y_i)$:

• $X = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]$
• $Y = [5, 7, 9, 11, 10, 9, 8, 7, 6, 5]$

Step 2: Calculate the means of $X$ and $Y$

The mean of $X$ is:

$\bar{X} = \frac{1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10}{10} = \frac{55}{10} = 5.5$

The mean of $Y$ is:

$\bar{Y} = \frac{5 + 7 + 9 + 11 + 10 + 9 + 8 + 7 + 6 + 5}{10} = \frac{77}{10} = 7.7$

Step 3: Calculate $(X_i - \bar{X})(Y_i - \bar{Y})$ for each pair

Step 4: Sum the products

$\sum (X_i - \bar{X})(Y_i - \bar{Y}) = 12.15 + 2.45 - 3.25 - 4.95 - 1.15 + 0.65 + 0.45 - 1.75 - 5.95 - 12.15 = -13.5$

Step 5: Calculate the covariance

Using the formula for covariance:

$\text{Cov}(X, Y) = \frac{1}{n} \sum (X_i - \bar{X})(Y_i - \bar{Y}) = \frac{-13.5}{10} = -1.35$

Conclusion:

The covariance of the given data set is -1.35.

Do you have any further questions or need more clarification on any part?

Here are some related questions you might want to explore:

1. What is the interpretation of negative covariance in this context?
2. How does covariance differ from correlation?
3. How would you calculate the variance of each variable separately?
4. Can covariance be normalized to always lie between -1 and 1?
5. How would adding a constant to each data point affect the covariance?

Tip: Covariance can give insight into the relationship between two variables, but it’s often helpful to also calculate the correlation coefficient to understand the strength and direction of the relationship.