To calculate the Pearson correlation coefficient $r$ for the given data, we follow these steps:

Given dataTo calculate the Pearson correlation coefficient $r$ for the given data, we follow these steps:

Given

• Number of TV commercials $x$: 5, 6, 13, 14, 19
• Car sales $y$ (in hundreds): 3, 2, 9, 5, 8

The Pearson correlation coefficient $r$ is calculated using the 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 (here $n = 5$),
• $\sum x$ is the sum of the $x$-values,
• $\sum y$ is the sum of the $y$-values,
• $\sum xy$ is the sum of the product of each corresponding $x$ and $y$-value,
• $\sum x^2$ is the sum of the squares of the $x$-values,
• $\sum y^2$ is the sum of the squares of the $y$-values.

Step 1: Compute the necessary sums

Let's calculate each sum:

$\sum x = 5 + 6 + 13 + 14 + 19 = 57$ $\sum y = 3 + 2 + 9 + 5 + 8 = 27$ $\sum xy = (5 \times 3) + (6 \times 2) + (13 \times 9) + (14 \times 5) + (19 \times 8) = 15 + 12 + 117 + 70 + 152 = 366$ $\sum x^2 = 5^2 + 6^2 + 13^2 + 14^2 + 19^2 = 25 + 36 + 169 + 196 + 361 = 787$ $\sum y^2 = 3^2 + 2^2 + 9^2 + 5^2 + 8^2 = 9 + 4 + 81 + 25 + 64 = 183$

Step 2: Plug the sums into the formula for $r$

Now, we substitute the values into the Pearson correlation coefficient formula:

$r = \frac{5 \times 366 - 57 \times 27}{\sqrt{(5 \times 787 - 57^2)(5 \times 183 - 27^2)}}$

First, calculate the numerator:

$5 \times 366 = 1830$ $57 \times 27 = 1539$ $1830 - 1539 = 291$

Next, calculate the denominator:

$5 \times 787 = 3935$ $57^2 = 3249$ $3935 - 3249 = 686$

$5 \times 183 = 915$ $27^2 = 729$ $915 - 729 = 186$

Now, calculate the square root part:

$\sqrt{686 \times 186} = \sqrt{127596} \approx 357.133286$

Finally, calculate $r$:

$r = \frac{291}{357.133286} \approx 0.815$

Final answer:

The Pearson correlation coefficient is approximately $r = 0.815$.

Would you like more details or have any questions?

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Here are five related questions:

1. What does a Pearson correlation coefficient of 0.815 signify about the relationship between TV commercials and car sales?
2. How can the correlation coefficient be interpreted in the context of marketing strategies?
3. What would a negative correlation coefficient indicate in this scenario?
4. How can one improve the accuracy of predictions based on the correlation coefficient?
5. What are some limitations of using Pearson's correlation in analyzing this data?

Tip: When interpreting the Pearson correlation, remember that correlation does not imply causation—it only measures the strength of a linear relationship between two variables.