The following table shows the age of customers and the amount of money they spent on
airtime in one month:
𝒙
Age
18 21 23 25 30 32 36 38 39 45
𝒚
Amount (R)
330 300 300 240 250 190 180 310 150 120
1.1 Determine the equation of the least squares regression line.

To determine the equation of the least squares regression line, we follow these steps:

### Step 1: Calculate the necessary sums

Given dataTo determine the equation of the least squares regression line, we follow these steps:

### Step 1: Calculate the necessary sums

Given 
- \( x \) (Age): \( 18, 21, 23, 25, 30, 32, 36, 38, 39, 45 \)
- \( y \) (Amount spent in R): \( 330, 300, 300, 240, 250, 190, 180, 310, 150, 120 \)

We need the following sums:
- \( \sum x \)
- \( \sum y \)
- \( \sum xy \)
- \( \sum x^2 \)

I'll calculate these sums.

\[
\sum x = 18 + 21 + 23 + 25 + 30 + 32 + 36 + 38 + 39 + 45 
\]

\[
\sum y = 330 + 300 + 300 + 240 + 250 + 190 + 180 + 310 + 150 + 120 
\]

\[
\sum xy = (18 \times 330) + (21 \times 300) + (23 \times 300) + (25 \times 240) + (30 \times 250) + (32 \times 190) + (36 \times 180) + (38 \times 310) + (39 \times 150) + (45 \times 120)
\]

\[
\sum x^2 = 18^2 + 21^2 + 23^2 + 25^2 + 30^2 + 32^2 + 36^2 + 38^2 + 39^2 + 45^2 
\]

Let's calculate these sums first.The sums are as follows:
- \( \sum x = 307 \)
- \( \sum y = 2370 \)
- \( \sum xy = 68230 \)
- \( \sum x^2 = 10129 \)

### Step 2: Calculate the slope (\( b \)) and intercept (\( a \)) of the regression line

The formulas for the slope (\( b \)) and intercept (\( a \)) are:

\[
b = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}
\]

\[
a = \frac{\sum y - b(\sum x)}{n}
\]

Where:
- \( n \) is the number of data points (in this case, \( n = 10 \)).

Let's calculate the slope \( b \) and intercept \( a \).The equation of the least squares regression line is:

\[
y = 434.47 - 6.43x
\]

This equation suggests that for each additional year of age, the amount spent on airtime decreases by approximately R6.43.

Would you like more details or have any questions about this process? 

Here are some related questions to explore further:

1. How do you interpret the slope in the context of this problem?
2. How can we use the regression equation to predict the amount spent for a given age?
3. What does the intercept represent in this context?
4. How can we assess the accuracy of this regression model?
5. How would the regression line change if the data were different?

**Tip:** The slope of a regression line indicates the direction and strength of the relationship between the variables.

Learn how to determine the equation of the least squares regression line using statistical methods. This problem involves calculating sums of data points, applying formulas for slope and intercept, and interpreting regression line equations. Suitable for advanced high school students studying statistics.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation