Question

Use technology to find the regression line to predict \(Y\) from \(X\). $$\begin{array}{lrlllll}\hline X & 10 & 20 & 30 & 40 & 50 & 60 \\\Y & 112 & 85 & 92 & 71 & 64 & 70 \\\\\hline\end{array}$$

Short answer

The exact regression line will depend on the specific results from the technology used, but it will come in the form of \(Y = aX + b\), where a is the slope and b is the Y-intercept. These values are calculated to minimize the square differences between the observed and predicted values of Y.

Step-by-step solution

List down the given data pairs

The given data pairs are (10,112), (20,85), (30,92), (40,71), (50,64) and (60,70).

Use technology to compute regression line

Use a graphing calculator or other software (like Excel, Google Sheets, or a statistical tool) to input the data. The regression line will be calculated by the software by minimizing the squared differences between the observed and predicted values of Y. The regression function is usually of the form \(Y = aX + b\), where a is the slope of the regression line and b is the Y-intercept.

Interpret the output

After running the regression in these software tools, you'll get the slope (a) and Y-intercept (b) values. They represent the equation of the regression line which best predicts Y based on X's values. Put these into the regression function \(Y = aX + b\).

Key concepts

Statistical Regression

In the realm of statistics, regression is a powerful tool used to describe the relationship between variables and forecast future observations. It essentially involves examining how a dependent variable, often denoted as Y, changes as one or more independent variables, indicated by X, vary.

When looking at a set of data points, such as those in the exercise with pairs like (10,112) and (20,85), statistical regression aims to find the best line—or curve—that fits these points. This 'best fit' line is calculated by minimizing the discrepancies between the actual data points and the ones predicted by the model.

For students to better understand and visualize this concept, plotting the data points on a graph is a helpful exercise. You can then draw a line through the data points, and it will show you the average direction of the data. Think of regression as trying to draw a straight line through a set of scattered points in a way that the distance of the points from the line is minimized overall. This process reveals trends and can inform predictions, which is invaluable not just in academics but also in real-world scenarios such as economics and engineering.

Linear Regression

Linear regression is a specific type of statistical regression that assumes a linear relationship between the independent variable (X) and the dependent variable (Y). This relationship is represented by a straight line with the equation of the form \( Y = aX + b \), where 'a' represents the slope of the line, indicating how much Y changes for a one-unit change in X, and 'b' is the Y-intercept, which is the value of Y when X is zero.

When employing linear regression to predict future values, software or a graphing calculator can perform the necessary calculations to find the optimal values for 'a' and 'b'. This is known as fitting the regression line to the data. The closer the data points are to the line, the better the predictive power of the line.

As illustrated in the textbook exercise, the goal is to find the line that best explains the variation in Y based on X. Hence, after using a technological tool to process the pairs of data, the output would give you the slope and intercept essential for building the regression equation. Remember that the context of the data is integral; in some cases, a visual inspection of the graph can also aid in discerning whether a linear model is appropriate or if another form of regression might be more suitable.

Data Analysis

Data analysis is an extensive field involving numerous techniques to scrutinize, transform, and model data with the aim of uncovering useful information, drawing conclusions, and supporting decision-making. In the context of our problem, data analysis involves collecting the given pairs of data, plotting them, and then employing statistical methods like regression to understand and explain the relationship between them.

Diligent data analysis often begins with a clear understanding of the data in question. In the case of our exercise, this means recognizing that each pair, such as (30,92), is a point that reflects a relationship between an independent variable (X) and a dependent variable (Y). Tools like spreadsheets or statistical software are then used to engage in sophisticated data analysis tasks, such as calculating regression equations and forecasting future points.

Improving one's ability in data analysis often includes mastering the art of recognizing patterns, understanding data collection methodologies, becoming proficient with computer-based analysis tools, and being capable of interpreting both numerical and graphical data to make informed predictions or decisions. It's essential to note that while tools and software are incredibly helpful, the validity and accuracy of any analysis rest upon the quality of the data and the appropriateness of the statistical techniques applied.