The regression of Price on Size of homes in Albuquerque had...

The regression of Price on Size of homes in Albuquerque had $R^{2}=71.4 \%,$ as described in Exercise $29 .$ a) What is the correlation between Size and Price? b) What would you predict about the Price of a home 1 SD above average in Size? c) What would you predict about the Price of a home 2 SDs below average in Size?

The regression of Price on Size of homes in Albuquerque had $R^{2}=71.4 \%,$ as described in Exercise $29 .$
a) What is the correlation between Size and Price?
b) What would you predict about the Price of a home 1 SD above average in Size?
c) What would you predict about the Price of a home 2 SDs below average in Size?

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Key Concepts

Standardized Regression Coefficient

When both the predictor and response variables are standardized (i.e., converted to z-scores with a mean of 0 and a standard deviation of 1), the slope of the regression line is equal to the correlation coefficient (r). This means that a change of 1 standard deviation in the predictor corresponds to a change of r standard deviations in the response.

Predicting Using Standard Deviations

In the context of regression, making predictions in terms of standard deviations involves using the standardized regression coefficient. Specifically, for any given deviation (in standard deviation units) of the predictor from its mean, the predicted response deviates from its mean by the product of that deviation and the correlation coefficient. This concept facilitates understanding the impact of operating in standardized units on the outcome variable.

Correlation Coefficient

The correlation coefficient (r) quantifies the strength and direction of the linear relationship between two variables. In simple linear regression, the square of the correlation coefficient (r²) equals the coefficient of determination (R²), which represents the proportion of variance in the response variable explained by the predictor variable.

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