In a regression equation applied to data in which the X and Y variables are both standardized (i.e., transformed into z scores), what is the value of the intercept?

Question

BlackTom AI · Accepted Answer

First, restate the scenario: we have a regression equation where both X and Y have been standardized to z-scores.

Next, recall the standardization effect: when a variable is transformed into a z-score, its mean becomes 0 and its standard deviation becomes 1.

In a simple linear regression of Z_Y on Z_X, the model has the form: Z_Y = b0 + b1 * Z_X, where Z_Y and Z_X are the standardized variables.

Consider the intercept b0 in this context. By definition of the regression equation, the expected value of Z_Y given Z_X is E[Z_Y | Z_X] = b0 + b1 * Z_X. The intercept b0 is the predicted value of Z_Y when Z_X = 0.

But since Z_X is a standardized variable with mean 0, the point Z_X = 0 corresponds to the mean of the original X transformed into a z-score. At this point, the expected value of Z_Y must equal the mean of Z_Y, which is 0 (because Z_Y is standardized to have mean 0). Therefore, E[Z_Y] = b0 + b1 * E[Z_X] = b0 + b1 * 0 = b0.

Since E[Z_Y] = 0, it follows that b0 = 0.

Intuitively, when both variables are centered and scaled, the regression line passes through the origin (0,0) in the standardized space, so the intercept is 0.

For completeness, note that the slope b1 in this standardized regression equals the correlation coefficient r between X and Y, but the key point here is that the intercept is determined by the centering to be zero, irrespective of the slope value.

If only one variable were standardized, the intercept would not necessarily be zero; the zero-intercept property arises specifically from standardizing both variables so that their means are zero.

PSYCH 3321 SP2025 (28481) Homework 9

In a regression equation applied to data in which the X and Y variables are both standardized (i.e., transformed into z scores), what is the value of the intercept?

View Explanation

Log in for full answers

Similar Questions

You have the following standardised regression model equation for a sample of data: ZŶi = 0.6ZXi Variable Y is “Daily Amount of Water Consumed” (in litres) Variable X is “Temperature” (in °C) Which of the following statements is true based on the standardised regression equation?

You have a sample of 20 individuals, and run a simple regression analysis using statistical software (with dependent variable Y, and independent variable X). One of the output tables is as follows: What is the standardised regression equation (to two decimal places)?

Refer to Scenario 8-1. You can see from the information in (1) that Charlie got a lot more education than the average person. When compared to the average person, the standardized regression model predicts that

In a regression equation applied to data in which the X and Y scores are standardized (i.e., transformed into z scores), what value of zY would you predict for a zX value of 0?

In a consumer society, many adults channel creativity into buying things

Economic stress and unpredictable times have resulted in a booming industry for self-help products

People born without creativity never can develop it

A product has a selling price of $20, a contribution margin ratio of 40% and fixed cost of $120,000. To make a profit of $30,000. The number of units that must be sold is: Type the number without $ and a comma. Eg: 20000

More Practical Tools for Students Powered by AI Study Helper

Homework AI Solver

Stylized AI Paper Writer

Plagiarism Checker Assistant

Citation AI Academic Writing Tool

In-Class Translation Assistant

AI Note Generator

AI Quiz Answers

Past Exam Questions from University Test Bank

Smart Practice Assistant

Adaptive Practice

Making Your Study Simpler