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Chapter 2: Problem 200

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. \(\mathrm{Hgt}=\) height in inches, Age \(=\) 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 for a 12-year-old child is obtained by substituting the age into the regression equation. The residual, or deviation from the regression prediction is obtained by subtracting the predicted height from the observed height. The slope indicates that for each extra year of age, the model predicts, on average, an increase in height of 2.74 inches. The intercept of 24.3 inches should represent a child's height at birth, which is not consistent with typical real-world observations.

Step by step solution

01

Calculate the Predicted Value

To find the predicted value (also known as the predicted height), substitute the age of the child into the regression equation. Here, the age of the child is given as 12 years. With the given regression equation \(\widehat{H g t}=24.3+2.74(\) Age \()), we have \(\widehat{H g t}=24.3+2.74(12)\)
02

Calculate the Residual

The residual is calculated as the observed data value minus the predicted data value. The observed height of the child (our data point) is 60 inches. To calculate the residual, perform the operation \(60 - \widehat{H g t}\)
03

Interpret the Slope

The slope of the regression equation, given by 2.74, is the estimated change in height for each additional year of age. That is, for each increase in age by 1 year, the predicted height increases by 2.74 inches, on average.
04

Interpret the Intercept

The intercept of the regression equation, given by 24.3, is the estimated value for height when the age of the child is 0 years. However, a child's height at birth is usually more than 24.3 inches. This signifies that the intercept might not make sense in the context of predicting a child's height because interpretation assumes you could have a child of age 0 years, which does not match with real-world understanding.

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Key Concepts

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

Predicted Value
In regression analysis, the predicted value is a crucial component that represents an estimate based on the regression equation. Essentially, it's what we expect the dependent variable to be, given a specific independent variable.
To calculate it, simply substitute the given value of the independent variable into the regression equation. In our exercise, the formula is \(\widehat{Hgt} = 24.3 + 2.74(\text{Age})\). For a child who is 12 years old, the predicted height is calculated by inserting 12 for the age, which results in:
  • \(\widehat{Hgt} = 24.3 + 2.74 \times 12\)
This simplifies to give the estimated height.
Predicted values are essential for forecasting outcomes and making comparisons with actual observed data.
Residual Calculation
Residuals are the differences between observed and predicted values. They provide insight into how accurate our predictions are. A residual can be found by subtracting the predicted value from the actual observed value. In doing so, we measure the discrepancy between these two.
Using the exercise's context where the observed height is 60 inches, the residual is expressed as:
  • Residual = Observed value - Predicted value \(= 60 - \widehat{Hgt}\)
This tells us how far our prediction was from the actual observation, showcasing the variability that is not explained by the regression model.
Smaller residuals suggest a better fit of the model to the data.
Interpreting Slope
The slope in a regression line indicates the relationship between the independent and dependent variables. Specifically, it reflects the rate of change in the dependent variable for every one-unit change in the independent variable.
In this example, the slope is 2.74, which means:
  • For each additional year of age, the child's height is expected to grow by 2.74 inches, on average.
The slope provides a clear understanding of how the variables interact, illustrating how significantly one variable influences the other. This information is critical when predicting future values or understanding the dynamics between the variables.
Interpreting Intercept
The intercept in regression indicates the expected value of the dependent variable when the independent variable is zero. In our model, the intercept is 24.3.
This suggests:
  • At age 0, the predicted height is 24.3 inches.
However, in practical terms, this may not reflect real-world conditions, as newborns are generally taller than 24.3 inches.
The intercept can sometimes seem unrealistic, especially when an independent variable value of zero is not possible in a real-world scenario, like age. Understanding intercepts involves recognizing these limitations and considering how values are best used within a sensible range in the model.

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Most popular questions from this chapter

Use the dataset BodyFat, which gives the percent of weight made up of body fat for 100 men as well as other variables such as Age, Weight (in pounds), Height (in inches), and circumference (in \(\mathrm{cm}\) ) measurements for the Neck, Chest, Abdomen, Ankle, Biceps, and Wrist. \(^{78}\) The regression line for predicting body fat percent using neck circumference is BodyFat \(=-47.9+1.75 \cdot\) Neck (a) What body fat percent does the line predict for a person with a neck circumference of \(35 \mathrm{~cm}\) ? Of \(40 \mathrm{~cm} ?\) (b) Interpret the slope of the line in context. (c) One of the men in the study had a neck circumference of \(38.7 \mathrm{~cm}\) and a body fat percent of \(11.3 .\) Find the residual for this man.

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Runs and Wins in Baseball In Exercise 2.150 on page \(104,\) we looked at the relationship between total hits by team in the 2014 season and division (NL or AL) in baseball. Two other variables in the BaseballHits dataset are the number of wins and the number of runs scored during the season. The dataset consists of values for each variable from all 30 MLB teams. From these data we calculate the regression line: \(\widehat{\text { Wins }}=34.85+0.070(\) Runs \()\) (a) Which is the explanatory and which is the response variable in this regression line? (b) Interpret the intercept and slope in context. (c) The San Francisco Giants won 88 games while scoring 665 runs in 2014. Predict the number of games won by San Francisco using the regression line. Calculate the residual. Were the Giants efficient at winning games with 665 runs?
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