To determine if there is a random relationship between the number of ads run (x) and the number of cars sold (y) for the given weeks, we can calculate the correlation coefficient (r). This will help determine the strength and direction of the linear relationship between the two variables.

The formula for the Pearson correlation coefficient $r$ is:

[ 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 (weeks in this case, which is 8),
• $\sum{x}$ is the sum of all the values of $x$,
• $\sum{y}$ is the sum of all the values of $y$,
• $\sum{xy}$ is the sum of the product of each pair of corresponding $x$ and $y$,
• $\sum{x^2}$ is the sum of the squares of all $x$ values,
• $\sum{y^2}$ is the sum of the squares of all $y$ values.

Given Data:

\hline \text{Week} & x (\text{Ads run}) & y (\text{Cars sold}) \\ \hline 1 & 8 & 10 \\ 2 & 15 & 22 \\ 3 & 20 & 26 \\ 4 & 13 & 24 \\ 5 & 25 & 24 \\ 6 & 16 & 20 \\ 7 & 18 & 23 \\ 8 & 22 & 25 \\ \hline \end{array}$$ Now, let’s calculate the necessary sums to compute $$r$$. - $$\sum{x} = 8 + 15 + 20 + 13 + 25 + 16 + 18 + 22 = 137$$ - $$\sum{y} = 10 + 22 + 26 + 24 + 24 + 20 + 23 + 25 = 174$$ - $$\sum{xy} = (8 \times 10) + (15 \times 22) + (20 \times 26) + (13 \times 24) + (25 \times 24) + (16 \times 20) + (18 \times 23) + (22 \times 25) = 80 + 330 + 520 + 312 + 600 + 320 + 414 + 550 = 3126$$ - $$\sum{x^2} = (8^2) + (15^2) + (20^2) + (13^2) + (25^2) + (16^2) + (18^2) + (22^2) = 64 + 225 + 400 + 169 + 625 + 256 + 324 + 484 = 2547$$ - $$\sum{y^2} = (10^2) + (22^2) + (26^2) + (24^2) + (24^2) + (20^2) + (23^2) + (25^2) = 100 + 484 + 676 + 576 + 576 + 400 + 529 + 625 = 3966$$ Now, substitute these values into the formula for the correlation coefficient $$r$$: \[ r = \frac{8 \times 3126 - 137 \times 174}{\sqrt{[8 \times 2547 - (137)^2][8 \times 3966 - (174)^2]}} \] Let’s compute this step-by-step. math # Given data n = 8 x = [8, 15, 20, 13, 25, 16, 18, 22] y = [10, 22, 26, 24, 24, 20, 23, 25] # Calculate sums sum_x = sum(x) sum_y = sum(y) sum_xy = sum(x[i] * y[i] for i in range(n)) sum_x2 = sum(xi**2 for xi in x) sum_y2 = sum(yi**2 for yi in y) # Calculate correlation coefficient numerator = n * sum_xy - sum_x * sum_y denominator = math.sqrt((n * sum_x2 - sum_x**2) * (n * sum_y2 - sum_y**2)) r = numerator / denominator rThe Pearson correlation coefficient $$r$$ is approximately **0.766**. This indicates a **moderate positive linear relationship** between the number of ads run and the number of cars sold. In other words, as the number of ads increases, the number of cars sold also tends to increase, but the relationship is not perfect. Would you like further details or an interpretation of these results? Here are 5 related questions for you to explore further: 1. How can we test the statistical significance of this correlation? 2. What does a correlation coefficient of 0.766 suggest about causality between ads and sales? 3. What are some other methods to analyze the relationship between these variables? 4. How do outliers affect the Pearson correlation coefficient? 5. How do we compute the regression line to predict sales based on ads? **Tip**: To ensure a reliable correlation, always check the data for linearity. If the relationship is nonlinear, Pearson's correlation may not give an accurate picture.