The following are the age​ (in years) and price​ (in hundreds of​ dollars) data for a certain type of car. A scatterplot of the data is given to the right.
0
2
4
6
8
10
200
300
400
500
x
y

A scatterplot has a horizontal x-axis labeled from 0 to 10 in increments of 1 and a vertical y-axis labeled from 200 to 500 in increments of 25. The following points are plotted: (6, 270); (7, 260); (6, 280); (1, 410); (1, 360); (4, 300); (4, 340); (4, 310); (1, 410); (5, 310). A line falls from left to right passing through the points (0, 415) and (5, 295). The points are clustered around the falling line. All coordinates are approximate.

Age​ (x)
6
7
6
1
1
4
4
4
1
5
Price​ (y)
269
256
275
405
364
298
336
309
409
306

Part 1
a. requals=enter your response here ​(Round to three decimal places as​ needed.)

To calculate the correlation coefficient \( r \), we need to follow these steps:

1. **Calculate the means of \( x \) (Age) and \( y \) (Price):**

   \[
   \bar{x} = \frac{6 + 7 + 6 + 1 + 1 + 4 + 4 + 4 + 1 + 5}{10}
   \]
   \[
   \bar{y} = \frac{269 + 256 + 275 + 405 + 364 + 298 + 336 + 309 + 409 + 306}{10}
   \]

2. **Compute the sums for each of the following:**

   - \( \sum (x_i - \bar{x})(y_i - \bar{y}) \)
   - \( \sum (x_i - \bar{x})^2 \)
   - \( \sum (y_i - \bar{y})^2 \)

3. **Use these sums to calculate \( r \):**

   \[
   r = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum (x_i - \bar{x})^2 \cdot \sum (y_i - \bar{y})^2}}
   \]

Let's go through these steps.

### Step 1: Calculate the means of \( x \) and \( y \)

\[
\bar{x} = \frac{6 + 7 + 6 + 1 + 1 + 4 + 4 + 4 + 1 + 5}{10} = \frac{39}{10} = 3.9
\]

\[
\bar{y} = \frac{269 + 256 + 275 + 405 + 364 + 298 + 336 + 309 + 409 + 306}{10} = \frac{3227}{10} = 322.7
\]

### Step 2: Compute the necessary sums

For each data point, calculate the following:

\[
(x_i - \bar{x})(y_i - \bar{y})
\]
\[
(x_i - \bar{x})^2
\]
\[
(y_i - \bar{y})^2
\]

Let's plug in the values:

\[
\text{for } x_1 = 6 \text{ and } y_1 = 269: 
(6 - 3.9)(269 - 322.7) = 2.1(-53.7) = -112.77
\]

Similarly, calculate for all the points.

\[
\sum (x_i - \bar{x})(y_i - \bar{y}) = \text{sum of all products calculated}
\]

### Step 3: Calculate \( r \)

Using the formula:

\[
r = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum (x_i - \bar{x})^2 \cdot \sum (y_i - \bar{y})^2}}
\]

I'll now compute the final value.The correlation coefficient \( r \) is approximately \(-0.957\) (rounded to three decimal places). This indicates a strong negative linear relationship between the age of the cars and their prices.

Would you like more details on this, or do you have any questions?

### Related Questions
1. How is the correlation coefficient interpreted in terms of strength and direction?
2. What does a negative correlation coefficient imply in this context?
3. How would the correlation change if there were an outlier in the data?
4. What are the limitations of using the correlation coefficient?
5. How would you calculate the line of best fit for this data?
6. Can you predict the price of a car given its age using the correlation coefficient?
7. What statistical test could you use to determine if the correlation is significant?
8. How does the correlation coefficient differ from the coefficient of determination \( r^2 \)?

**Tip:** When calculating correlation, remember that it only measures linear relationships. Non-linear patterns require different statistical approaches.

Learn how to calculate the correlation coefficient for age and price data of cars. This problem involves statistical analysis, including calculating means, sums, and applying the correlation coefficient formula. Suitable for undergraduate students and data analysts.

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