Find the correlation coefficient, r, of the data described below.
An automobile dealership ran a radio ad campaign to help sell its least popular model. The dealership always used the same ad but broadcast it a different number of times each week.
The sales manager tallied the number of ads broadcast each week, x, and the weekly sales of the model, y.
Ads	Sales
5	1
8	6
9	19
14	14
18	2
Round your answer to the nearest thousandth.

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.

Find the correlation coefficient, r, of the data described below. An automobile dealership ran a radio ad campaign to help sell its least popular model. The dealership always used the same ad but broadcast it a different number of times each week. The sales manager tallied the number of ads broadcast each week, x, and the weekly sales of the model, y. Ads Sales 5 1 8 6 9 19 14 14 18 2. Round your answer to the nearest thousandth.

Discover how to calculate the correlation coefficient r for an automobile dealership's radio ad campaign data. With weekly ads ranging from 5 to 18 and corresponding sales from 1 to 19, we provide a step-by-step solution to understand the relationship between advertising and sales. Suitable for Grades 10-12.

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