Given hypothetical data on weekly family consumption expenditure (y) and weekly income (x), we are tasked with finding:

1. The values of $a$ (the intercept) and $b$ (the slope) for the regression equation of the form $y = a + bx$.
2. Fitting the regression line based on this equation.

Step-by-Step Solution:

The linear regression equation is: $y = a + bx$ Where:

• $y$ is the dependent variable (weekly family consumption expenditure),
• $x$ is the independent variable (weekly family income),
• $a$ is the intercept, and
• $b$ is the slope of the regression line.

To calculate $a$ and $b$:

• The slope $b$ is given by: $b = \frac{n \sum xy - \sum x \sum y}{n \sum x^2 - (\sum x)^2}$
• The intercept $a$ is given by: $a = \frac{\sum y - b \sum x}{n}$ Where:
• $n$ is the number of data points,
• $\sum x$ is the sum of all $x$-values (income),
• $\sum y$ is the sum of all $y$-values (expenditure),
• $\sum xy$ is the sum of the product of corresponding $x$ and $y$-values,
• $\sum x^2$ is the sum of the squares of the $x$-values.

Let's now calculate the necessary values for $a$ and $b$.

The data points are:

Income (x): 80, 100, 120, 140, 160, 180, 200, 220, 240, 260

Expenditure (y): 70, 65, 90, 95, 110, 115, 120, 140, 155, 150

Let me calculate the values of $a$ and $b$.The values of the intercept and slope for the regression line are:

• Intercept ($a$): $24.45$
• Slope ($b$): $0.51$

(ii) Fitting the regression line:

The regression equation is: $y = 24.45 + 0.51x$

This line represents the relationship between weekly income ($x$) and weekly family consumption expenditure ($y$).

Would you like more details or clarification on any step?

Here are some follow-up questions:

1. How can we interpret the slope in this context?
2. What does the intercept represent in real-world terms?
3. How would the regression line change if the slope was negative?
4. Can you predict the expenditure for an income of $300 using this regression model?
5. What statistical methods can be used to check the goodness of fit?

Tip: In regression analysis, always check the residuals to assess the quality of the fit.