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Chapter 2: Problem 187

In Exercises 2.187 to 2.190 , two variables are defined, a regression equation is given, and one data point is given. (a) Find the predicted value for the data point and compute the residual. (b) Interpret the slope in context. (c) Interpret the intercept in context, and if the intercept makes no sense in this context, explain why. \(H g t=\) height in inches, \(A g e=\) age in years of a child \(\widehat{H g t}=24.3+2.74(\) Age \()\); data point is a child 12 years old who is 60 inches tall

Short Answer

Expert verified
The predicted height of the child is 57.68 inches and the residual is 2.32 inches. The slope suggests that for every one year increase in a child's age, the height increases by 2.74 inches. The intercept represents the predicted height of a newborn child but it does not make sense in this context as it does not align with typical newborn height.

Step by step solution

01

Calculate Predicted Value

Plug in the value of age (12) into the regression equation \( \widehat{H g t}=24.3+2.74( Age )\) to find the predicted height, we get: \( \widehat{H g t}=24.3+2.74( 12 )=57.68 \) inches
02

Calculate the Residual

The formula for residual is observed value - predicted value. Here, the observed value is 60 inches and the predicted value is 57.68 inches. So, the residual is: \( 60 - 57.68 = 2.32 \) inches
03

Interpret the Slope

The slope of the regression line is \(2.74\). This implies that for each one-year increase in age, we predict the height to increase by \(2.74\) inches.
04

Interpret the Intercept

The intercept of the regression line is \(24.3\). This indicates the predicted height of a child when the age is 0. However, this might not make much practical sense since the height of a newborn child is typically more than \(24.3\) inches. Therefore, the intercept does not seem to have meaningful interpretation in this context.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Regression Equation
A regression equation provides a mathematical way to model the relationship between an independent variable and a dependent variable. It's typically represented as a linear equation. In this exercise, the regression equation is given by:\[ \widehat{Hgt} = 24.3 + 2.74(Age) \]Here, "\(\widehat{Hgt}\)" is the predicted height, and "Age" is the independent variable. The numbers 24.3 and 2.74 are known as the intercept and the slope of the equation, respectively.
  • The intercept (24.3) represents the expected value of the dependent variable (height) when the independent variable (age) is zero. Although, in real life, newborns don't have a height of 24.3 inches.
  • The slope (2.74) indicates the rate of change; for each additional year of age, the predicted height increases by 2.74 inches.
Understanding this equation allows us to predict the height of a child at different ages by simply plugging in the age value into the equation.
Predicted Value
The predicted value is what we get when we substitute a specific number into our regression equation to forecast the outcome. For instance, by inserting the age of 12 years into the given regression equation:\[ \widehat{Hgt} = 24.3 + 2.74 \times 12 \]we compute that the predicted height is: \[ \widehat{Hgt} = 24.3 + 32.88 = 57.68 \text{ inches} \]
  • This means that, according to the regression model, a 12-year-old child is expected to be about 57.68 inches tall.
Predicted values help us understand what outcomes to expect based on known inputs, and they form a vital part of regression analysis in forecasting and modeling scenarios.
Residual Calculation
In regression analysis, the residual is the difference between the observed value and the predicted value from the regression model. It tells us how much the prediction deviates from the actual measurement.To calculate the residual for the child in this exercise, you first find the predicted height using the regression equation. We've found that it's 57.68 inches. The observed height is 60 inches.Residual = Observed Height - Predicted Height \[= 60 - 57.68 = 2.32 \text{ inches} \]
  • The positive residual of 2.32 inches indicates that the actual height is greater than what we predicted.
  • Residuals are crucial for assessing the accuracy of a regression model, as smaller residuals generally mean better model fit.
Residuals help identify how well a model represents reality, highlighting the discrepancies between expected and observed outcomes.

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