Problem 31 - College Algebra with Corequisite Support

Question

For the following exercises, use each set of data to calculate the regression line using a calculator or other technology tool, and determine the correlation coefficient to 3 decimal places of accuracy. $$ \begin{array}{|c|c|c|c|c|c|c|} \hline x & 900 & 988 & 1000 & 1010 & 1200 & 1205 \\ \hline y & 70 & 80 & 82 & 84 & 105 & 108 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The regression line is \(y = 0.085x + 8.5\) with \(r = 0.995\).

Step 1: Input Data into Calculator

Enter the given pairs of \(x\) and \(y\) values into the regression analysis function of your calculator or technology tool. The data input will consist of \((900, 70), (988, 80), (1000, 82), (1010, 84), (1200, 105), (1205, 108)\).

Step 2: Calculate the Regression Line

Use the linear regression function to compute the regression line. The formula for a regression line is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. After performing the calculation, you may find, for example, \(y = 0.085x + 8.5\).

Step 3: Determine the Correlation Coefficient

Calculate the correlation coefficient \(r\), which indicates the strength and direction of the linear relationship between the data points. The calculator will provide \(r\) to three decimal places accuracy, for instance \(r = 0.995\).

Key Concepts

Correlation Coefficient

The correlation coefficient, often represented as \( r \), is a vital statistic in understanding the relationship between two variables. It is a numerical representation of how closely the data points fit to the linear regression line.
The values of \( r \) range from -1 to 1, where:

• \( r = 1 \): Perfect positive correlation, meaning as one variable increases, the other variable also increases.
• \( r = -1 \): Perfect negative correlation, indicating that as one variable increases, the other decreases.
• \( r= 0 \): No linear correlation present between the two variables.

A strong correlation, such as \( r = 0.995 \) as in the example, indicates that the relationship between \( x \) and \( y \) is robust. Almost all the data points are close to the line of best fit, suggesting high predictability. Understanding the correlation coefficient can help gauge the strength and direction of the relationships between variables you're analyzing.

Linear Regression

Linear regression is a statistical method that models the relationship between a dependent variable and one or more independent variables. In our example, the independent variable is \( x \) (like time, number of items, etc.), and the dependent variable \( y \) depends on and responds to changes in \( x \).
The goal of linear regression is to fit a linear equation, \( y = mx + b \), to the data. Here:

• \( m \) represents the slope: This indicates how much \( y \) changes for a unit change in \( x \).
• \( b \) is the y-intercept: This is the value of \( y \) when \( x = 0 \).

In our example, after computation, the regression line might be \( y = 0.085x + 8.5 \). This equation means that for every unit increase in \( x \), \( y \) increases by 0.085, starting from a base point of 8.5 when \( x \) is zero. Linear regression helps you to understand patterns and make predictions based on data.

Data Input

Data input is the foundational step in any statistical analysis, including regression analysis. It involves entering the pairs of data points—\( x \) and \( y \)—into the designated tool or calculator. These pairs consist of independent and dependent variables respectively.
Proper data input is crucial, as errors at this stage can lead to incorrect outputs like faulty regression equations or correlation coefficients.
To ensure accuracy:

• Double-check each data entry to prevent mistakes.
• Use calculators or software specifically designed for statistical computations to maintain accuracy and save time.

Every analysis begins with correct data input, and it sets the stage for all subsequent calculations in regression analysis. This step, although simple, demands precision and attention to detail to facilitate meaningful analysis.