Problem 3 - Advanced Engineering Mathematics

Question

Find the least squares line for the given data. $$ (1,1),(2,1.5),(3,3),(4,4.5),(5,5) $$

Step-by-Step Solution

Answer

The least squares line is \( y = 1.3x - 0.9 \).

Step 1: Calculate the means

First, calculate the mean of the x-values and the mean of the y-values. For the x-values (1, 2, 3, 4, 5), the mean is calculated as \( \bar{x} = \frac{1 + 2 + 3 + 4 + 5}{5} = 3 \). For the y-values (1, 1.5, 3, 4.5, 5), the mean is \( \bar{y} = \frac{1 + 1.5 + 3 + 4.5 + 5}{5} = 3 \).

Step 2: Calculate the slope (m)

To find the slope \( m \) of the least squares line, use the formula \( m = \frac{\sum{(x_i - \bar{x})(y_i - \bar{y})}}{\sum{(x_i - \bar{x})^2}} \). Substitute the values: \( m = \frac{((1-3)(1-3)+ (2-3)(1.5-3)+ (3-3)(3-3)+ (4-3)(4.5-3)+ (5-3)(5-3))}{((1-3)^2 + (2-3)^2 + (3-3)^2 + (4-3)^2 + (5-3)^2)} = \frac{13}{10} = 1.3 \).

Step 3: Calculate the y-intercept (b)

Now calculate the y-intercept \( b \) using the formula \( b = \bar{y} - m\bar{x} \). Since both \( \bar{y} \) and \( \bar{x} \) are 3, we have \( b = 3 - (1.3)(3) = -0.9 \).

Step 4: Write the equation of the line

Now that you have \( m = 1.3 \) and \( b = -0.9 \), you can write the least squares line in the form \( y = mx + b \). Thus, the equation is \( y = 1.3x - 0.9 \).

Key Concepts

Slope Calculation

The calculation of slope is a fundamental step in finding the least squares regression line.
The slope essentially tells us how steep the line is. It shows how much the dependent variable (often labeled as "y") changes for every one-unit change in the independent variable ("x").To find the slope \(m\) of the least squares line, we use the formula:

• \( m = \frac{\sum{(x_i - \bar{x})(y_i - \bar{y})}}{\sum{(x_i - \bar{x})^2}} \)

This formula might look complex, but let's break it down:

• \( \bar{x} \: \) is the mean of the x-values.
• \( \bar{y} \: \) is the mean of the y-values.
• \( x_i \: \) and \( y_i \: \) are the individual data points.
• The numerator \( (x_i - \bar{x})(y_i - \bar{y}) \: \) is the sum of the products of deviations of each point from the means.
• The denominator \( (x_i - \bar{x})^2 \: \) is the sum of the squared deviations of the x-values from their mean.

In our example, the slope came out to be 1.3. This means that for every one-unit increase in x, y increases by 1.3 units.

Y-Intercept Calculation

After calculating the slope, the next step is to find the y-intercept of the least squares regression line.
The y-intercept is the point where the line crosses the y-axis. This is important: it shows the value of y when x is zero.To find the y-intercept \(b\), we use the formula:

• \( b = \bar{y} - m\bar{x} \)

Where:

• \( \bar{y} \: \) is the mean of the y-values.
• \( m \: \) is the slope you calculated.
• \( \bar{x} \: \) is the mean of the x-values.

In this scenario, both \( \bar{y} \) and \( \bar{x} \) are 3, and the slope \( m \) is 1.3. Plugging these values into the formula gives us:

• \( b = 3 - (1.3)(3) = -0.9 \)

So, the y-intercept is -0.9. This indicates the expected value of y when x is zero is -0.9, based on our data.

Data Analysis

Data analysis plays a crucial role in understanding and interpreting the results of the least squares regression.
It allows us to make predictions based on the relationship between variables detected through our regression line.Here's why data analysis is important:

• **Prediction**: The regression line helps predict the dependent variable based on new or future values of the independent variable.
• **Trend Detection**: By analyzing the slope and intercept, we can detect trends in the data over time.
• **Quantifying Relationships**: The precise numerical output (our slope and intercept) quantifies how tightly our variables are related.
• **Decision Making**: Insights gathered from regression and data analysis guide decisions in business, economics, science, and more.

Understanding these data aspects ensures better interpretations and supports using statistical results effectively. In our specific problem, by knowing \( y = 1.3x - 0.9 \), we gain a clear picture of how changes in x influence y, aiding data-driven decisions.