Problem 31 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the least-squares line (Problem 30 ) for the data \((3,2),(4,3),(5,4),(6,4)\), and \((7,5)\).

Step-by-Step Solution

Answer

The least-squares line is \(y = 0.56x + 0.8\).

Step 1: Calculate the means of x and y

First, we find the mean of the x-values and the mean of the y-values.The x-values are 3, 4, 5, 6, 7. \(\bar{x} = \frac{3 + 4 + 5 + 6 + 7}{5} = 5\).The y-values are 2, 3, 4, 4, 5. \(\bar{y} = \frac{2 + 3 + 4 + 4 + 5}{5} = 3.6\).

Step 2: Calculate the slope (m)

Use the formula for the slope of the least-squares line: \[ m = \frac{\sum{(x_i - \bar{x})(y_i - \bar{y})}}{\sum{(x_i - \bar{x})^2}}.\] Substituting the values we have:\((3 - 5)(2 - 3.6) + (4 - 5)(3 - 3.6) + (5 - 5)(4 - 3.6) + (6 - 5)(4 - 3.6) + (7 - 5)(5 - 3.6) = 5.6\).\((3 - 5)^2 + (4 - 5)^2 + (5 - 5)^2 + (6 - 5)^2 + (7 - 5)^2 = 10\).Thus, \(m = \frac{5.6}{10} = 0.56\).

Step 3: Calculate the y-intercept (b)

Use the formula for the y-intercept: \(b = \bar{y} - m\bar{x}\).Substituting the values: \( b = 3.6 - 0.56 \times 5 = 3.6 - 2.8 = 0.8\).

Step 4: Write the equation of the least-squares line

Combine the slope and y-intercept to write the equation.The equation of the least-squares line is \(y = 0.56x + 0.8\).

Key Concepts

Linear Regression

Linear regression is a fundamental technique used in statistics to understand the relationship between two variables by fitting a linear equation to observed data. The main aim is to predict the dependent variable (usually denoted as y) based on the independent variable (x). The linear equation is typically in the form of \(y = mx + b\), where \(m\) is the slope of the line and \(b\) is the y-intercept.
In simple linear regression:

• The slope \(m\) indicates how much y changes for a unit change in x.
• The y-intercept \(b\) represents the value of y when x is zero.

Linear regression is widely used due to its simplicity and interpretability. It provides a straightforward way to assess the strength of the relationship between variables and is foundational for more advanced statistical techniques.

Statistical Analysis

Statistical analysis in the context of linear regression involves examining data using statistical methods to establish relationships. It helps in identifying patterns and trends within the data to make informed predictions and decisions.
Statistical analysis encompasses various steps:

• Data Collection: Gathering data points for the variables of interest.
• Descriptive Statistics: Calculating mean, median, and other statistics to summarize the data.
• Inferential Statistics: Using statistical tests to make predictions or infer trends from the data.

For linear regression problems, statistical analysis often includes calculating the correlation coefficient, which measures the strength and direction of the linear relationship. Additionally, statistical tests like hypothesis testing are used to confirm whether there is a significant relationship between the variables.

Data Fitting

Data fitting is the process of constructing a function that best fits a series of data points. In the context of linear regression, the least-squares method is commonly employed to derive the best-fitting line.
The least-squares method aims to minimize the sum of the squares of the residuals, which are the differences between the observed and predicted values. This method ensures:

• The fitted line is as close as possible to all the data points.
• The overall error in fitting the data is minimized.

The least-squares line provides a way to visually represent the trend in the data while offering a predictive model for making future predictions. It's crucial in statistics for model construction and evaluation, ensuring that the chosen model accurately reflects the data.