To find the correlation coefficient, $r$, for the given data, we can use the formula:

$r = \frac{n \sum (xy) - \sum x \sum y}{\sqrt{ \left( n \sum x^2 - (\sum x)^2 \right) \left( n \sum y^2 - (\sum y)^2 \right) }}$

where:

• $n$ is the number of data points,
• $x$ and $y$ are the given values,
• $\sum x$ is the sum of $x$ values,
• $\sum y$ is the sum of $y$ values,
• $\sum xy$ is the sum of the products of $x$ and $y$ for each data point,
• $\sum x^2$ is the sum of the squares of $x$,
• $\sum y^2$ is the sum of the squares of $y$.

Let's calculate each component step by step.

Step 1: Calculate the Sums

Using the data provided:

$x$	$y$	$x \cdot y$	$x^2$	$y^2$
5	1	5	25	1
8	6	48	64	36
9	19	171	81	361
14	14	196	196	196
18	2	36	324	4

Now, summing each column:

• $\sum x = 5 + 8 + 9 + 14 + 18 = 54$
• $\sum y = 1 + 6 + 19 + 14 + 2 = 42$
• $\sum (xy) = 5 + 48 + 171 + 196 + 36 = 456$
• $\sum x^2 = 25 + 64 + 81 + 196 + 324 = 690$
• $\sum y^2 = 1 + 36 + 361 + 196 + 4 = 598$

Step 2: Substitute Values into the Formula

We have:

• $n = 5$,
• $\sum x = 54$,
• $\sum y = 42$,
• $\sum (xy) = 456$,
• $\sum x^2 = 690$,
• $\sum y^2 = 598$.

Now, substituting these into the correlation coefficient formula:

$r = \frac{5 \cdot 456 - 54 \cdot 42}{\sqrt{ \left( 5 \cdot 690 - 54^2 \right) \left( 5 \cdot 598 - 42^2 \right) }}$

Step 3: Perform the Calculations

1. Calculate the numerator: $5 \cdot 456 = 2280, \quad 54 \cdot 42 = 2268$ $\text{Numerator} = 2280 - 2268 = 12$

2. Calculate the denominator:

   • For $\sum x^2$ part: $5 \cdot 690 = 3450, \quad 54^2 = 2916$ $5 \cdot 690 - 54^2 = 3450 - 2916 = 534$
   • For $\sum y^2$ part: $5 \cdot 598 = 2990, \quad 42^2 = 1764$ $5 \cdot 598 - 42^2 = 2990 - 1764 = 1226$

   So, the denominator is: $\sqrt{534 \cdot 1226} = \sqrt{654684} \approx 809.12$

3. Calculate $r$: $r = \frac{12}{809.12} \approx 0.015$

Final Answer

The correlation coefficient $r$ is approximately:

$r \approx 0.015$

This indicates a very weak positive correlation between the number of ads and weekly sales, suggesting little to no linear relationship.

Would you like more details or have any questions?

Here are some additional related questions:

1. What does a low correlation coefficient imply about the data?
2. How could we interpret a negative correlation coefficient if it were the case?
3. How can the data be visualized to better understand the correlation?
4. Are there other methods for assessing relationships besides correlation?
5. What would a perfect correlation look like in this context?

Tip: For a better understanding of data relationships, try visualizing the data with a scatter plot before calculating correlation.