Problem 3 - A Modern Introduction to Probability and Statistics: Understanding Why and How

Question

Suppose we have the following bivariate dataset: \((1,3.1) \quad(1.7,3.9) \quad(2.1,3.8) \quad(2.5,4.7) \quad(2.7,4.5)\). a. Determine the least squares estimates \(\hat{\alpha}\) and \(\hat{\beta}\) of the parameters of the regression line \(y=\alpha+\beta x\). You may use that \(\sum x_{i}=10, \sum y_{i}=20\), \(\sum x_{i}^{2}=21.84\), and \(\sum x_{i} y_{i}=41.61\). b. Draw in one figure the scatterplot of the data and the estimated regression line \(y=\hat{\alpha}+\hat{\beta} x\).

Step-by-Step Solution

Answer

The least squares estimates are \( \hat{\alpha} = 2.25 \) and \( \hat{\beta} = 0.875 \). The regression line is \( y = 2.25 + 0.875x \).

Step 1: Calculate Slope Estimate

To find the least squares estimates for the regression line, we'll start by calculating \( \hat{\beta} \), the slope, using the formula:\[ \hat{\beta} = \frac{n(\sum x_i y_i) - (\sum x_i)(\sum y_i)}{n(\sum x_i^2) - (\sum x_i)^2} \]Given:- \( n = 5 \)- \( \sum x_i = 10 \)- \( \sum x_i^2 = 21.84 \)- \( \sum y_i = 20 \)- \( \sum x_i y_i = 41.61 \)Plugging in these values:\[ \hat{\beta} = \frac{5(41.61) - (10)(20)}{5(21.84) - (10)^2} = \frac{208.05 - 200}{109.2 - 100} = \frac{8.05}{9.2} \approx 0.875 \]

Step 2: Calculate Intercept Estimate

Next, calculate \( \hat{\alpha} \), the y-intercept, using the formula:\[ \hat{\alpha} = \frac{\sum y_i}{n} - \hat{\beta} \cdot \frac{\sum x_i}{n} \]Using the given sums and \( \hat{\beta} \approx 0.875 \), we have:\[ \hat{\alpha} = \frac{20}{5} - 0.875 \cdot \frac{10}{5} = 4 - 1.75 = 2.25 \]

Step 3: Write Regression Equation

With both \( \hat{\beta} \) and \( \hat{\alpha} \) calculated, the equation of the least squares regression line is:\[ y = 2.25 + 0.875x \]

Step 4: Draw the Scatterplot and Regression Line

To visualize the dataset and the regression line, plot the data points \((1, 3.1), (1.7, 3.9), (2.1, 3.8), (2.5, 4.7), (2.7, 4.5)\) on a graph. Then, draw the line \( y = 2.25 + 0.875x \) by using any two x-values to calculate corresponding y-values on this line.- Example points on the regression line using \( x = 1 \) and \( x = 3 \): - When \( x = 1 \), \( y = 2.25 + 0.875 \cdot 1 = 3.125 \) - When \( x = 3 \), \( y = 2.25 + 0.875 \cdot 3 = 4.875 \)This line should be drawn such that it minimizes the vertical distances from all given data points to the line itself.

Key Concepts

Least Squares Method

The Least Squares Method is a statistical approach used to determine the equation of the best fit line that minimizes the sum of the squares of the vertical distances of the data points from the line. This method helps us find the most accurate linear relationship between a dependent and an independent variable.

To understand the process, consider a set of bivariate data, which represents paired observations. Our goal is to model a line given by the equation \( y = \alpha + \beta x \), where \( \alpha \) is the y-intercept and \( \beta \) is the slope.
For the calculation, the slope \( \hat{\beta} \) is derived from the formula:

• \( \hat{\beta} = \frac{n(\sum x_i y_i) - (\sum x_i)(\sum y_i)}{n(\sum x_i^2) - (\sum x_i)^2} \)

This formula accounts for the sum of the products of \( x \) and \( y \), the squares of \( x \), and the sums of individual \( x \) and \( y \) values.

Afterward, use the intercept formula to find \( \hat{\alpha} \):

• \( \hat{\alpha} = \frac{\sum y_i}{n} - \hat{\beta} \cdot \frac{\sum x_i}{n} \)

Calculating these values will give you the regression equation, providing insights into the relationship within the data.

Bivariate Data

Bivariate data consists of pairs of linked numerical observations. It is the cornerstone of examining relationships between two different variables, aiming to understand how the variation in one variable corresponds to variation in the other.

To work with bivariate data effectively, consider each pair \((x_i, y_i)\) as a data point on a coordinate axis. For instance, the data points \((1, 3.1), (1.7, 3.9), (2.1, 3.8), (2.5, 4.7), (2.7, 4.5)\) from the exercise provide a visual representation of potential linear relationships. This kind of data is essential for regression analysis since it enables us to find systematic patterns of association between the variables.

Bivariate data analysis aims to develop models like regression lines, which predict the response of the dependent variable \( y \) from changes in the independent variable \( x \). This analysis helps to understand trends, make predictions, and identify anomalies within the data set.

Scatterplot

A scatterplot is a type of graph used to display bivariate data. Each point on a scatterplot represents a pair of values, providing a visual representation of the relationship between two variables.

To create a scatterplot, plot each pair of data points on an axis. For example, if you have the bivariate data points such as \((1, 3.1), (1.7, 3.9), (2.1, 3.8), (2.5, 4.7), (2.7, 4.5)\), each pair corresponds to a point on the graph.

Scatterplots are beneficial because:

• They visually reveal the relationship or correlation between the variables.
• They help us see potential outliers or unusual data points.
• They provide a basis for further statistical modeling, like drawing a regression line.

The regression line can be drawn through a scatterplot to highlight the linear relationship. The objective of this line is to best fit through all the plotted points by minimizing the vertical differences, helping to make predictions from the independent variable.