Question

The formula for a regression equation is \(Y^{\prime}=2 X+9\). a. What would be the predicted score for a person scoring 6 on \(\mathrm{X} ?\) b. If someone's predicted score was \(14,\) what was this person's score on \(X ?\)

Short answer

a. 21 b. 2.5

Step-by-step solution

Identify the Given Formula

The formula for the regression equation is given as \(Y^{\prime}=2X+9\). This equation can be used to solve for either \(Y^{\prime}\) when \(X\) is known or to find \(X\) when \(Y^{\prime}\) is known.

Calculate Predicted Score for Given X

For part a of the problem, we need to find \(Y^{\prime}\) when \(X = 6\). Substituting 6 in place of \(X\) in the equation: \(Y^{\prime} = 2(6) + 9 = 12 + 9 = 21\). Thus, the predicted score is 21.

Solve for X when Predicted Score is Known

For part b, we have \(Y^{\prime} = 14\) and need to find the corresponding \(X\). Start with the equation \(14 = 2X + 9\). Subtract 9 from both sides to get \(5 = 2X\), and then divide both sides by 2 to isolate \(X\). Therefore, \(X = 2.5\).

Key concepts

Regression Equation

A regression equation is a mathematical expression used in statistics to describe the relationship between two variables. Often expressed in the form of a line, it helps predict the value of a dependent variable based on the value of an independent variable. At its core, a regression equation represents a straight line:

• The form is typically written as: \( Y^{\prime} = aX + b \)
• Where \( Y^{\prime} \) is the predicted score or output, \( X \) is the known input value, \( a \) is the slope, and \( b \) is the y-intercept.

In our example, \( Y^{\prime} = 2X + 9 \), the slope \( (2) \) indicates how much \( Y^{\prime} \) changes with every one-unit increase in \( X \). The y-intercept \( (9) \) describes where the line crosses the y-axis when \( X = 0 \). Templates like this are foundational in regression analysis to make predictions or understand trends.

Predicted Score

Calculating a predicted score involves using the regression equation to find the value of the dependent variable based on a given \( X \). By substituting a specific number into the formula, we can predict or estimate outcomes.
Here's how it's done:

• Take the regression equation: \( Y^{\prime} = 2X + 9 \).
• If \( X = 6 \), substitute it into the equation: \( Y^{\prime} = 2(6) + 9 \).
• Calculate the result: \( Y^{\prime} = 12 + 9 = 21 \).

Thus, when someone scores a 6 on \( X \), the predicted score for \( Y^{\prime} \) is 21.
This process highlights how you can anticipate an outcome, helping in various fields such as academic assessments or business forecasting.

Solving for Variables

Solving for variables involves using algebraic manipulation to find a specific unknown. When we know \( Y^{\prime} \) and want to find \( X \), we rearrange the regression equation. This is a crucial skill in regression analysis when the output is known, but the initial input is unknown. Here’s how you can solve for \( X \):

• Start with the equation: \( 14 = 2X + 9 \).
• Subtract 9 from both sides to isolate the term with \( X \): \( 5 = 2X \).
• Divide each side by 2 to solve for \( X \): \( X = \frac{5}{2} = 2.5 \).

This exemplifies how understanding and rearranging equations allows you to backtrack and uncover initial values.
Mastering these steps is key in mathematics and situations where the original inputs must be deduced from an outcome.