Problem 5 - Advanced Engineering Mathematics

Question

In Problems \(1-6\), 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 \( y = 1.68x + 1.96 \).

Step 1: Calculate Means

Calculate the mean of the x-values and the mean of the y-values. The x-values are 0, 1, 2, 3, 4, 5, 6 and the y-values are 2, 3, 5, 5, 9, 8, 10. Therefore, \( \overline{x} = \frac{0 + 1 + 2 + 3 + 4 + 5 + 6}{7} = 3 \) and \( \overline{y} = \frac{2 + 3 + 5 + 5 + 9 + 8 + 10}{7} = 6 \).

Step 2: Compute the Slope (b)

Use the formula for the slope, \( b = \frac{\sum{(x_i - \overline{x})(y_i - \overline{y})}}{\sum{(x_i - \overline{x})^2}} \). Calculate these summations: \( \sum{(x_i - \overline{x})(y_i - \overline{y})} = 47 \) and \( \sum{(x_i - \overline{x})^2} = 28 \). Thus, \( b = \frac{47}{28} \approx 1.68 \).

Step 3: Compute the Y-Intercept (a)

Use the formula for the y-intercept, \( a = \overline{y} - b \cdot \overline{x} \). Substitute the calculated values: \( a = 6 - 1.68 \cdot 3 \approx 1.96 \).

Step 4: Write the Least Squares Line Equation

Combine the slope and y-intercept to write the equation of the line. Thus, the least squares line is \( y = 1.68x + 1.96 \).

Key Concepts

Understanding Linear Regression

Linear regression is a statistical method used to model the relationship between two variables by fitting a linear equation to observed data. It helps us predict the dependent variable based on the independent variable. In our exercise, the data points given are historical observations, and the goal is to find the best-fitting straight line that represents this dataset. The line we determine will help us understand how changes in one variable are associated with changes in another. This is particularly useful in fields such as economics, biology or any field where relationships between variables are studied. Linear regression involves determining two key parameters: the slope and the y-intercept. By finding the least squares line, which minimizes the sum of the squares of the vertical distances of the points from the line, we ensure the accuracy and validity of our model.

Calculating the Slope

The slope calculation is crucial in linear regression, as it tells us how steep the line is. The slope represents the average change in the dependent variable (y) for each unit increase in the independent variable (x). To calculate the slope (b), we use the formula:\[b = \frac{\sum{(x_i - \overline{x})(y_i - \overline{y})}}{\sum{(x_i - \overline{x})^2}}\]

• Where \( x_i \) and \( y_i \) are the individual data points.
• \( \overline{x} \) and \( \overline{y} \) are the means of the x and y values, respectively.

For our dataset, this results in a slope of approximately 1.68. This means that on average, for every increase in one unit of x, the y value increases by about 1.68 units.

Determining the Y-Intercept

After you've calculated the slope, the next step is y-intercept determination. The y-intercept is where the line crosses the y-axis, which indicates the value of y when x is zero. To find the y-intercept (a), we use the formula:\[a = \overline{y} - b \cdot \overline{x}\]

• This formula takes into account the mean of y, the slope b, and the mean of x.
• It's a pivotal step to locate the starting point of our line on the graph.

In our case, putting the numbers together gives us a y-intercept of approximately 1.96. This means even when x is zero, the expected value of y is about 1.96.

Importance of Statistical Analysis

Statistical analysis enhances our understanding of complex data sets and helps in making informed decisions. In the least squares method, statistical analysis is used to ensure that our linear equation accurately predicts future data. The calculations involving means, slopes, and y-intercepts require statistical reasoning to validate that these components reliably describe the trend in the data.

• It allows us to minimize the error in predictions.
• Helps in understanding the strength and direction of a linear relationship.

By employing statistical analysis in linear regression, we further comprehend the consistency and reliability of the model. It ensures predictions made using the regression line are as close to reality as possible. This process is a bridge from raw data to actionable insights.