To calculate Karl Pearson's coefficient of correlation ($r$) between advertising expenditure and sales from the data provided in the image, we will follow these steps:

1. Tabulate Data:

   • List the values of $x$ (Advertising Expenditure) and $y$ (Sales).
   • Compute the mean of $x$ ($\bar{x}$) and $y$ ($\bar{y}$).
   • Calculate $(x - \bar{x})$, $(y - \bar{y})$, $(x - \bar{x})^2$, $(y - \bar{y})^2$, and $(x - \bar{x})(y - \bar{y})$.

2. Formula for Correlation Coefficient: $r = \frac{\sum{(x - \bar{x})(y - \bar{y})}}{\sqrt{\sum{(x - \bar{x})^2} \cdot \sum{(y - \bar{y})^2}}}$

3. Plug in values and solve.

Let me compute the values for you. numpy as np

Given data

advertising_expenditure = np.array([12, 15, 18, 23, 24, 38, 42, 48]) # in '000 Tk sales = np.array([5.0, 5.6, 5.8, 7.0, 7.2, 8.8, 9.2, 9.5]) # in Lakh Tk

Calculate means

mean_x = np.mean(advertising_expenditure) mean_y = np.mean(sales)

Calculate deviations from mean

x_deviation = advertising_expenditure - mean_x y_deviation = sales - mean_y

Calculate squares and product of deviations

x_deviation_squared = x_deviation2 y_deviation_squared = y_deviation2 xy_deviation_product = x_deviation * y_deviation

Calculate sums

sum_x_deviation_squared = np.sum(x_deviation_squared) sum_y_deviation_squared = np.sum(y_deviation_squared) sum_xy_deviation_product = np.sum(xy_deviation_product)

Calculate correlation coefficient (r)

r = sum_xy_deviation_product / np.sqrt(sum_x_deviation_squared * sum_y_deviation_squared)

mean_x, mean_y, sum_x_deviation_squared, sum_y_deviation_squared, sum_xy_deviation_product, rThe calculations yield the following:

• Mean of advertising expenditure ($\bar{x}$) = 27.5 ('000 Tk)
• Mean of sales ($\bar{y}$) = 7.2625 (Lakh Tk)
• Sum of squared deviations for $x$: $\sum{(x - \bar{x})^2} = 1260.0$
• Sum of squared deviations for $y$: $\sum{(y - \bar{y})^2} = 21.21875$
• Sum of product of deviations: $\sum{(x - \bar{x})(y - \bar{y})} = 161.25$

Karl Pearson's coefficient of correlation ($r$) is: $r = 0.9862$

This indicates a very strong positive correlation between advertising expenditure and sales.

Would you like me to provide detailed step-by-step calculations or further interpretation of the result?

5 Questions You May Want to Explore:

1. How do we interpret a correlation coefficient near $+1$?
2. Can causation between advertising and sales be inferred from this correlation?
3. What are the potential limitations of using Pearson's $r$ in this context?
4. How would the result change if one of the data points was an outlier?
5. What other statistical methods could be used to analyze this relationship?

Tip:

Remember, correlation does not imply causation. Even with a strong $r$, external factors could influence sales.