Question

Based on the table below, compute the regression line that predicts \(Y\) from \(X\). $$ \begin{array}{|c|c|c|c|c|} \hline \mathbf{M}_{\mathrm{x}} & \mathrm{M}_{\mathrm{Y}} & \mathrm{s}_{\mathrm{x}} & \mathrm{S}_{\mathrm{Y}} & \mathrm{r} \\ \hline 10 & 12 & 2.5 & 3.0 & -0.6 \\ \hline \end{array} $$

Short answer

The equation of the regression line is \(Y = 19.2 - 0.72X\).

Step-by-step solution

Understand the Data Provided

The table provides several important statistical measures: the mean of \(X\) variables (\(M_x = 10\)), the mean of \(Y\) variables (\(M_Y = 12\)), the standard deviation of \(X\) variables (\(s_x = 2.5\)), the standard deviation of \(Y\) variables (\(S_Y = 3.0\)), and the correlation coefficient (\(r = -0.6\)). Our task is to find the equation of the regression line that predicts \(Y\) based on \(X\).

Calculate the Slope of the Regression Line

The slope \(b\) of the regression line can be calculated using the formula:\[ b = r \times \left( \frac{S_Y}{s_x} \right) \]Substitute the values provided:\[ b = -0.6 \times \left( \frac{3.0}{2.5} \right) = -0.6 \times 1.2 = -0.72 \]

Calculate the Intercept of the Regression Line

The intercept \(a\) of the regression line can be found using the formula:\[ a = M_Y - b \times M_x \]Substitute the values of \(M_Y\), \(b\), and \(M_x\):\[ a = 12 - (-0.72) \times 10 = 12 + 7.2 = 19.2 \]

Write the Equation of the Regression Line

Now that we have both the slope \(b\) and the intercept \(a\), we can write the equation of the regression line:\[ Y = a + bX \]Substitute the values for \(a\) and \(b\):\[ Y = 19.2 - 0.72X \]

Key concepts

Correlation Coefficient

The correlation coefficient, often represented by the symbol \( r \), is a statistical measure that represents the strength and direction of a linear relationship between two variables. It ranges from -1 to 1.
- A coefficient of 1 indicates a perfect positive linear relationship, meaning as one variable increases, the other increases proportionally.- A coefficient of -1 indicates a perfect negative linear relationship, meaning as one variable increases, the other decreases proportionally.- A coefficient of 0 suggests no linear relationship between the variables.
In the example provided, the correlation coefficient is \( r = -0.6 \). This implies a moderate negative correlation between \( X \) and \( Y \), indicating that as \( X \) increases, \( Y \) tends to decrease. Understanding the correlation coefficient helps predict how changes in one variable might affect another.

Statistical Measures

Statistical measures such as the mean and standard deviation provide a summary of the data's central tendency and dispersion, respectively. The mean is the average value of a dataset and gives us a sense of the central point.- The mean of \( X \) values ( \( M_x \)) is 10.- The mean of \( Y \) values ( \( M_Y \)) is 12.
These values are essential in the calculation of the regression line because they help establish the line's central point.
Statistical measures are critical for determining how data points are spread out from the mean. They also help in understanding the variability in datasets, which is crucial for accurately predicting relationships between variables.

Standard Deviation

Standard deviation is a statistical measure that quantifies the amount of variation or dispersion of a set of values. A lower standard deviation means that the values tend to be closer to the mean of the set, while a higher standard deviation indicates that the values are spread out over a wider range.
- The standard deviation for \( X \) ( \( s_x \)) is given as 2.5.- The standard deviation for \( Y \) ( \( S_Y \)) is given as 3.0.
Standard deviation is crucial in the context of regression analysis because it is used to calculate the slope of the regression line, indicating how much \( Y \) changes for a unit change in \( X \). Understanding standard deviation helps in interpreting the reliability and variability of predictions made by the regression model.

Regression Line Equation

The regression line equation is a crucial concept in regression analysis as it defines the relationship between the predicted variable \( Y \) and the independent variable \( X \). The general formula of a regression line is:
\[ Y = a + bX \]
where \( a \) is the intercept and \( b \) is the slope of the line.
- The intercept \( a = 19.2 \) indicates where the line crosses the Y-axis when \( X \) is zero.- The slope \( b = -0.72 \) indicates the change in \( Y \) for each unit change in \( X \).
This equation is derived using statistical measures such as the mean and standard deviation, as well as the correlation coefficient. Understanding the regression line equation allows us to predict the value of \( Y \) for any given \( X \), which is a primary goal of regression analysis.