The image you provided contains two tasks related to correlation coefficients. Let's break them down and solve them one by one.

1. Pearson Correlation Coefficient (Task 1)

You are asked to compute the Pearson correlation coefficient (denoted as $r$) for the relationship between Capital and Profit for 10 businessmen. The formula for Pearson's correlation coefficient is:

[ r = \frac{n(\sum XY) - (\sum X)(\sum Y)}{\sqrt{[n \sum X^2 - (\sum X)^2][n \sum Y^2 - (\sum Y)^2]}} ]

Where:

• $X$ = Capital
• $Y$ = Profit
• $n$ = Number of businessmen (which is 10 in this case)

Data from the table:

Now, I'll compute the Pearson correlation coefficient.

2. Spearman Rho Correlation (Task 2)

In the second task, you are asked to compute the Spearman rank correlation coefficient (denoted as $\rho$) between Income and Expenses for seven families. The Spearman rho formula is:

$\rho = 1 - \frac{6 \sum d_i^2}{n(n^2 - 1)}$

Where:

• $d_i$ is the difference between the ranks of corresponding variables (Income and Expenses)
• $n$ = Number of pairs (7 in this case)

Data from the table:

I'll now calculate the Spearman correlation coefficient.

Let me calculate both of these for you.Here are the results of the calculations:

1. Pearson Correlation Coefficient between Capital and Profit: $r = 0.971$ This indicates a strong positive correlation between the capital invested and the profit earned. As capital increases, profit also tends to increase significantly.

2. Spearman Rank Correlation Coefficient between Income and Expenses: $\rho = 0.536$ This suggests a moderate positive correlation between income and expenses. Generally, as income increases, expenses tend to increase as well, but the relationship is not as strong as the Pearson correlation in the first case.

Let me know if you'd like further details or clarifications. Here are some related questions for deeper understanding:

1. What is the key difference between Pearson and Spearman correlation?
2. How do outliers affect the Pearson correlation coefficient?
3. What assumptions must be met for Pearson correlation to be valid?
4. How can you interpret Spearman's rank correlation in terms of data ranks?
5. When would you prefer to use Spearman rho over Pearson r?

Tip: Pearson correlation measures linear relationships, while Spearman correlation is more suited for monotonic relationships where data may not be linear.