Problem 5 - Advanced Engineering Mathematics

Question

Find the least squares line for the given data. $$ (0,2),(1,3),(2,5),(3,5),(4,9),(5,8),(6,10) $$

Step-by-Step Solution

Answer

The least squares line is \( y = 2.14x - 0.42 \).

Step 1: Calculate Means

First, we calculate the mean of the x-values and y-values. The x-values are 0, 1, 2, 3, 4, 5, and 6. Their mean is \( \bar{x} = \frac{0+1+2+3+4+5+6}{7} = 3 \) . The y-values are 2, 3, 5, 5, 9, 8, and 10. Their mean is \( \bar{y} = \frac{2+3+5+5+9+8+10}{7} = 6 \).

Step 2: Compute the Slope

Now, we calculate the slope \( m \) of the least squares line using the formula \( m = \frac{\sum_{i=1}^{n}(x_i - \bar{x})(y_i - \bar{y})}{\sum_{i=1}^{n}(x_i - \bar{x})^2} \). Here's how the calculations work out:- \( \sum (x_i - \bar{x})(y_i - \bar{y}) = (0-3)(2-6) + (1-3)(3-6) + (2-3)(5-6) + (3-3)(5-6) + (4-3)(9-6) + (5-3)(8-6) + (6-3)(10-6) = 60 \)- \( \sum (x_i - \bar{x})^2 = (0-3)^2 + (1-3)^2 + (2-3)^2 + (3-3)^2 + (4-3)^2 + (5-3)^2 + (6-3)^2 = 28 \)Thus, the slope \( m = \frac{60}{28} \approx 2.14 \).

Step 3: Compute the Intercept

Next, we find the y-intercept \( b \) using the formula \( b = \bar{y} - m \cdot \bar{x} \). Substitute the previously found values: \( b = 6 - 2.14 \cdot 3 = -0.42 \).

Step 4: Formulate the Least Squares Line Equation

With the values of \( m \) and \( b \) established, we can write the equation of the least squares line as \( y = 2.14x - 0.42 \).

Key Concepts

Slope Calculation

In least squares regression, calculating the slope is a crucial step to understand how changes in one variable impact another. For our given dataset, we first computed the mean of the x-values and y-values. These are fundamental to finding the slope.The slope, represented as \( m \), reveals the rate of change between the variables. To find this, we use the formula: \[ m = \frac{\sum(x_i - \bar{x})(y_i - \bar{y})}{\sum(x_i - \bar{x})^2} \] This equation helps us determine the best-fit line for the data. In simpler terms, the numerator represents how the x and y values vary together, while the denominator accounts for how much the x values vary individually.For the dataset provided, when substituting in, we have: - \( \sum (x_i - \bar{x})(y_i - \bar{y}) = 60 \) - \( \sum (x_i - \bar{x})^2 = 28 \) Thus, the slope calculation results in \[ m = \frac{60}{28} \approx 2.14 \] This slope of approximately 2.14 suggests that for every unit increase in x, y increases by about 2.14 units.

Y-Intercept Calculation

Once we have the slope, the next step is to determine the y-intercept, denoted as \( b \). The y-intercept is the point where the line crosses the y-axis, providing a starting point for the line on the graph.The formula used for this is: \[ b = \bar{y} - m \cdot \bar{x} \] This allows us to relate the slope we've calculated to the actual points in our dataset.For our data:- \( \bar{x} = 3 \) and \( \bar{y} = 6 \)- Using our slope \( m = 2.14 \), the calculation becomes: \( b = 6 - 2.14 \times 3 = -0.42 \) The y-intercept of \(-0.42\) implies that when \( x = 0 \), the predicted \( y \) value on this line is \(-0.42\). This value helps define the starting point of our regression line in the real-world context of the data.

Data Analysis

Analyzing data using least squares regression is a powerful method to understand relationships between variables. With our line equation \( y = 2.14x - 0.42 \), we can predict y values from given x values. This analytical approach is valuable in making predictions or understanding trends. Several key aspects should be considered when performing data analysis using least squares regression:

• **Line of Best Fit**: The least squares method aims to minimize the differences between the data points and the predicted values given by the regression line.
• **Predictive Power**: It allows us to make forecasts based on the model, which can guide decisions and strategies.
• **Correlation Insight**: The slope indicates the strength and direction of a linear relationship between variables. In our case, a slope of 2.14 suggests a moderately strong positive link between x and y.

The insights derived from such analysis can lead to better decision-making and richer understanding of the data in various disciplines, from economics to social sciences. By understanding the components of the least squares line, one can grasp the dynamics between correlated variables more clearly.