Question

Identifying a Model and\({R^2}\)Different samples are collected, and each sample consists of IQ scores of 25 statistics students. Let x represent the standard deviation of the 25 IQ scores in a sample, and let y represent the variance of the 25 IQ scores in a sample. What formula best describes the relationship between x and y? Which of the five models describes this relationship? What should be the value of\({R^2}\)?

Short answer

\(y = {x^2}\)is the equation that would best describe the relationship between x (standard deviation of the IQ scores) and y (variance of the IQ scores).

Out of the five non-linear models, the quadratic model is the most appropriate.

The value of \({R^2}\) should be approximately equal to 1.

Step-by-step solution

Given information

A regression equation is to be computed where the response variable is the variance of IQ scores of students, and the predictor variable is the standard deviation of the IQ scores.

The sample size is 25.

Relation of variance and standard deviation

If s is the sample standard deviation of a sample, then the sample variance is equal to\({s^2}\).

Mathematically, the sample variance of a variable is the square of the sample standard deviation.

Here, x denotes the standard deviation of the IQ scores, and y represents the variance of the IQ scores.

Thus, the following equation would best describe the relation between x and y:

\(y = {x^2}\)

Type of non-linear model

There are five types of non-linear models that are commonly used:

Linear Model with the general formula:\(y = a + b{x^2}\)

Logarithmic Model with the general formula:\(y = a + b\ln \left( x \right)\)

Power Model with the general formula:\(y = a{x^b}\)

Quadratic Model with the general formula:\(y = a{x^2} + bx + c\)

Exponential Model with the general formula: \(y = a{b^x}\)

For the given relation, the model's equation resembles the quadratic model \(y = a{x^2} + bx + c\) where a =1, b=0 and c=0. Thus, the most appropriate model for the given relation \(y = {x^2}\)is the quadratic model.

Value of \({R^2}\)

\({R^2}\)indicates how good the constructed model describes the relationship.

It lies between 0 and 1.

\({R^2}\)equal to 1 implies that the fitted model perfectly represents the relation between the given variables.

The actual relation between sample variance and sample standard deviation is that the sample variance is the square of the sample standard deviation.

Since the regression model\(y = {x^2}\)truly represents the actual relationship,the value of the\({R^2}\)should be equal to 1.