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Problem 2

AMC 10/12, early questions

Algebra Difficulty 3.0 Find the answer

The regression equation $\hat{y}=0.85x-82.71$ is used to predict the weight of female college students based on their height, where $x$ is in centimeters (cm) and $\hat{y}$ is in kilograms (kg). Calculate the residual for an individual with the measurements $(160,53)$ .

Your answer

Official solution

Analysis

This question primarily examines the definition and calculation of residuals, requiring an understanding of the formula for calculating residuals. It's quite basic.

To find the residual, calculate the difference between the predicted value and the actual value.

Solution

Given the regression equation $\hat{y}=0.85x-82.71$ , when $x=160$ , we have $\hat{y}=0.85 \times 160 - 82.71 = 53.29$ .

Therefore, the residual for the individual $(160,53)$ is $53 - 53.29 = -0.29$ .

Hence, the answer is $\boxed{-0.29}$ .

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.