To fit a regression equation to the data and predict the sales revenue when units sold are 133, we need to perform the following steps:

Data:

• Sales revenue (in Rs. lakhs): 61, 42, 43, 40, 102, 80, 90
• Units sold: 75, 66, 65, 65, 85, 71, 75

Step 1: Organize the data in a tabular form:

Units Sold	Sales Revenue (Rs. Lakhs)
75	61
66	42
65	43
65	40
85	102
71	80
75	90

Step 2: Calculate the regression equation

We will fit a linear regression model of the form:

$\text{Sales Revenue} = \beta_0 + \beta_1 \times \text{Units Sold}$

Where:

• $\beta_0$ is the intercept
• $\beta_1$ is the slope of the line

The formula to calculate the slope ($\beta_1$) and intercept ($\beta_0$) is:

$\beta_1 = \frac{n \sum x_i y_i - \sum x_i \sum y_i}{n \sum x_i^2 - (\sum x_i)^2}$ $\beta_0 = \frac{\sum y_i - \beta_1 \sum x_i}{n}$

Where:

• $x_i$ are the units sold (independent variable),
• $y_i$ are the sales revenue (dependent variable),
• $n$ is the number of data points.

Step 3: Apply the formulas to the data.

I will now calculate the regression equation and predict the sales revenue when the units sold are 133. numpy as np

Data

units_sold = np.array([75, 66, 65, 65, 85, 71, 75]) sales_revenue = np.array([61, 42, 43, 40, 102, 80, 90])

Calculate required sums

n = len(units_sold) sum_x = np.sum(units_sold) sum_y = np.sum(sales_revenue) sum_xy = np.sum(units_sold * sales_revenue) sum_x2 = np.sum(units_sold**2)

Calculate the slope (beta1) and intercept (beta0)

beta1 = (n * sum_xy - sum_x * sum_y) / (n * sum_x2 - sum_x**2) beta0 = (sum_y - beta1 * sum_x) / n

Regression equation: Sales Revenue = beta0 + beta1 * Units Sold

Now predict the sales revenue when units sold is 133

units_sold_prediction = 133 sales_revenue_prediction = beta0 + beta1 * units_sold_prediction

(beta0, beta1, sales_revenue_prediction)The regression equation fitted to the given data is:

$\text{Sales Revenue} = -156.98 + 3.10 \times \text{Units Sold}$

Using this equation, when 133 units are sold, the predicted sales revenue is approximately Rs. 255.50 lakhs.

Would you like further details on how the regression equation is derived or have any additional questions? Here are some related questions you may find interesting:

1. How can I interpret the slope and intercept of a regression line?
2. What are the assumptions made in linear regression?
3. How do we assess the goodness of fit for this regression model?
4. What are alternative methods to predict sales revenue based on units sold?
5. How would the sales revenue change if units sold increased to 150?

Tip: Linear regression works best when the relationship between the variables is approximately linear. Always check the residuals (the difference between predicted and actual values) to ensure the model is appropriate.