To compute the sample covariance between two sets of data, we can use the following formula:

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

Where:

• $X_i$ and $Y_i$ are the individual data points from the two sets.
• $\bar{X}$ and $\bar{Y}$ are the means of the sets $X$ and $Y$.
• $n$ is the number of items in the sample (in this case, $n = 11$).

Steps:

1. Compute the means $\bar{X}$ and $\bar{Y}$.
2. Calculate the deviations from the mean for each data point.
3. Multiply the deviations for corresponding $X_i$ and $Y_i$ values.
4. Sum the products and divide by $n - 1$.

Let’s compute this step-by-step.

Data:

• $X: 4, 12, 18, 13, 7, 20, 9, 14, 2, 17, 1$
• $Y: 12, 36, 54, 39, 21, 60, 27, 42, 6, 51, 3$

First, we'll compute the means of $X$ and $Y$.

$\bar{X} = \frac{1}{n} \sum_{i=1}^{n} X_i, \quad \bar{Y} = \frac{1}{n} \sum_{i=1}^{n} Y_i$

Let’s compute these values.The means of the datasets are:

• $\bar{X} = 10.636$ (rounded to three decimal places)
• $\bar{Y} = 31.909$ (rounded to three decimal places)

Next, we'll compute the sample covariance using the deviations from these means. The formula we'll use is:

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

I'll calculate this now.The sample covariance between the two sets of data is $128.564$ (rounded to three decimal places).

Would you like any further details or explanations on this calculation?

Here are some related questions you might find useful:

1. How is sample covariance different from population covariance?
2. What does a positive covariance indicate about the relationship between two variables?
3. How does covariance relate to correlation?
4. Can you explain the steps involved in calculating covariance manually?
5. How would covariance change if the data were scaled?

Tip: Covariance helps determine the direction of the relationship between two variables, but not the strength. For the strength, correlation is more informative.