To calculate the correlation coefficient $r$ between the price and the number of bids, we can use the following formula:

[ 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:

• $n$ is the number of data points (in this case, 5),
• $x$ is the number of bids,
• $y$ is the price in dollars.

Given data:

Price (y)	Number of Bids (x)
23	12
32	13
35	15
41	16
44	19

Step-by-step solution:

1. Calculate the sums:

• $\sum{x} = 12 + 13 + 15 + 16 + 19 = 75$
• $\sum{y} = 23 + 32 + 35 + 41 + 44 = 175$
• $\sum{xy} = (23)(12) + (32)(13) + (35)(15) + (41)(16) + (44)(19)$
• $\sum{x^2} = 12^2 + 13^2 + 15^2 + 16^2 + 19^2$
• $\sum{y^2} = 23^2 + 32^2 + 35^2 + 41^2 + 44^2$

Let’s compute these:

$\sum{xy} = (23 \times 12) + (32 \times 13) + (35 \times 15) + (41 \times 16) + (44 \times 19) = 276 + 416 + 525 + 656 + 836 = 2709$

$\sum{x^2} = 12^2 + 13^2 + 15^2 + 16^2 + 19^2 = 144 + 169 + 225 + 256 + 361 = 1155$

$\sum{y^2} = 23^2 + 32^2 + 35^2 + 41^2 + 44^2 = 529 + 1024 + 1225 + 1681 + 1936 = 5395$

2. Apply the correlation coefficient formula:

Substitute the values into the formula:

[ r = \frac{5 \times 2709 - 75 \times 175}{\sqrt{[5 \times 1155 - (75)^2] [5 \times 5395 - (175)^2]}} ]

Let’s calculate each part:

$5 \times 2709 = 13545,\quad 75 \times 175 = 13125$ $5 \times 1155 = 5775,\quad 75^2 = 5625$ $5 \times 5395 = 26975,\quad 175^2 = 30625$

Now, plug these values into the equation:

$r = \frac{13545 - 13125}{\sqrt{(5775 - 5625)(26975 - 30625)}}$

Simplifying:

$r = \frac{420}{\sqrt{(150)(-3640)}}$

The negative value in the square root indicates an error in the calculation process, possibly due to data handling. Let me recalculate these steps to provide the accurate correlation coefficient value.

I'll get back to you with the refined answer in a moment! numpy as np

Data

x = np.array([12, 13, 15, 16, 19]) # Number of Bids y = np.array([23, 32, 35, 41, 44]) # Price in Dollars

Calculate the correlation coefficient

correlation = np.corrcoef(x, y)[0, 1] round(correlation, 3)The correlation coefficient $r$ between the number of bids and the item's selling price is approximately 0.933.

This indicates a strong positive correlation, meaning that as the number of bids increases, the selling price tends to increase as well.

Would you like more details on this calculation or any further clarification? Here are some related questions you might find interesting:

1. How does the correlation coefficient help in predicting relationships between variables?
2. What are some limitations of using the correlation coefficient in data analysis?
3. What is the difference between correlation and causation?
4. How can we interpret a negative correlation coefficient?
5. What other statistical methods could be used to analyze this data?

Tip: When calculating correlation, it's important to visualize the data using a scatter plot first. This helps in confirming the nature of the relationship (linear, non-linear, etc.).