Hypothetical data on weekly family consumption expenditure (y) and weekly income (x).

$

70

65

90

95

110

115

120

140

155

150

$

80

100

120

140

160

180

200

220

240

260

required

(i)                  find the value of a and b?

(ii)                fit the regression line?

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.

Hypothetical data on weekly family consumption expenditure (y) and weekly income (x). Required (i) find the value of a and b, (ii) fit the regression line.

Explore how to find the slope and intercept of a linear regression model using hypothetical data on weekly income and family consumption expenditure. This problem involves applying the least squares regression method with step-by-step calculations. Suitable for advanced high school or college students.

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