Researchers wondered whether the size of a person's brain...

Researchers wondered whether the size of a person's brain was related to the individual's mental capacity. They selected a sample of right-handed introductory psychology students who had SAT scores higher than $1350 .$ The subjects were administered the Wechsler (1981) Adult Intelligence Scale-Revised exam to obtain their IQ scores. MRI scans, performed at the same facility, consisted of 18 horizontal magnetic resonance images. The computer counted all pixels with nonzero gray scale in each of the 18 images, and the total count served as an index for brain size. The resulting data are presented in the table. (TABLE CAN'T COPY) (a) Find the least-squares regression line treating MRI count as the explanatory variable and IQ as the response variable. (b) What do you notice about the value of the slope? Why does this result seem reasonable based on the scatter diagram and linear correlation coefficient obtained in Problem 31 of Section $4.1 ?$ (c) When there is no relation between the explanatory and response variable, we use the mean value of the response variable, $\bar{y},$ to predict. Predict the IQ of an individual whose MRI count is $1,000,000 .$ Predict the IQ of an individual whose MRI count is $830,000 .$

Researchers wondered whether the size of a person's brain was related to the individual's mental capacity. They selected a sample of right-handed introductory psychology students who had SAT scores higher than $1350 .$ The subjects were administered the Wechsler (1981) Adult Intelligence Scale-Revised exam to obtain their IQ scores. MRI scans, performed at the same facility, consisted of 18 horizontal magnetic resonance images. The computer counted all pixels with nonzero gray scale in each of the 18 images, and the total count served as an index for brain size. The resulting data are presented in the table. (TABLE CAN'T COPY)
(a) Find the least-squares regression line treating MRI count as the explanatory variable and IQ as the response variable.
(b) What do you notice about the value of the slope? Why does this result seem reasonable based on the scatter diagram and linear correlation coefficient obtained in Problem 31 of Section $4.1 ?$
(c) When there is no relation between the explanatory and response variable, we use the mean value of the response variable, $\bar{y},$ to predict. Predict the IQ of an individual whose MRI count is $1,000,000 .$ Predict the IQ of an individual whose MRI count is $830,000 .$

Key Concepts

Least-Squares Regression

This method finds the best-fitting line through a set of data points by minimizing the sum of the squares of the vertical distances (residuals) between the observed responses and the values predicted by the line. It is a fundamental technique in linear regression analysis for estimating the relationship between variables.

Explanatory and Response Variables

In regression analysis, the explanatory (or independent) variable is the one used to predict or explain changes in another variable, while the response (or dependent) variable is the outcome of interest. Understanding and correctly identifying these roles is essential for building an appropriate regression model.

Interpretation of Slope in Linear Regression

The slope of the regression line represents the expected change in the response variable for a one-unit increase in the explanatory variable. This concept is crucial because it quantifies the strength and direction of the linear relationship between the variables.

Scatter Diagram and Correlation Coefficient

A scatter diagram is a graphical representation of the relationship between two quantitative variables, helping to visualize potential linear trends. The correlation coefficient quantifies the strength and direction of this relationship, and is closely linked to the slope of the regression line, providing an overall measure of linear association.

Prediction Using Regression and the Mean

Regression models are commonly used to predict the value of a response variable based on specific values of the explanatory variable. However, when there is no significant linear relationship between the variables, the mean of the response variable is often used as the best predictor. This concept underscores the importance of assessing model validity before making predictions.

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