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Chapter 2: Problem 201

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. \(B A C=\) blood alcohol content (\% of alcohol in the blood), Drinks \(=\) number of alcoholic drinks. \(\widehat{B A C}=-0.0127+0.018(\) Drinks \() ;\) data point is an individual who consumed 3 drinks and had a \(B A C\) of 0.08.

Short Answer

Expert verified
The predicted BAC for an individual who consumed 3 drinks is 0.041. The residual is 0.039. The slope shows that for each additional drink consumed, the predicted BAC increases by 0.018. The intercept, -0.0127, isn't meaningful in this context because BAC can't be negative, especially when no drinks are consumed.

Step by step solution

01

Calculate the Predicted Blood Alcohol Content (BAC)

Using the given regression equation, the predicted BAC (\(\widehat{B A C}\)) for an individual who has consumed 3 drinks can be calculated as follows: \(\widehat{B A C}= -0.0127 + 0.018*3 = 0.041\)
02

Compute the Residual

The residual is calculated by subtracting the predicted BAC from the observed BAC. In this case, the observed BAC is 0.08, and the predicted BAC from the regression equation is 0.041. Therefore, the residual = 0.08 - 0.041 = 0.039
03

Interpret the Slope

In this context, the slope (0.018) of the regression equation represents the predicted increase in blood alcohol content (BAC) for each additional alcoholic drink consumed. This means that for every additional drink consumed, the BAC is predicted to increase by 0.018.
04

Interpret the Intercept

The intercept (-0.0127) of the regression equation represents the predicted BAC when no drinks are consumed. Since BAC cannot be less than 0 (especially when no drinks are consumed), the intercept is not meaningful in this context. If it were positive, it would indicate the predicted BAC with zero alcohol consumption, but a negative BAC doesn't make sense, showing that the regression model doesn't fit the data perfectly at zero drinks.

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Key Concepts

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

Predicted Value Calculation
When we perform a linear regression analysis, one of the first steps is often to calculate a predicted value for a given data point. This helps us understand how well our model fits the data. In our example, we were given a regression equation: \[\widehat{BAC} = -0.0127 + 0.018 \times \text{Drinks}\]To find the predicted blood alcohol content (BAC) for an individual who consumed 3 drinks, we plug in 3 for Drinks in our equation:\[\widehat{BAC} = -0.0127 + 0.018 \times 3\]This results in a predicted BAC of 0.041.
  • **Why is this important?** By predicting expected values, we can assess how well the model describes the relationship between independent and dependent variables.
  • **Application:** This calculation helps in projecting outcomes under various input scenarios, like estimating someone's BAC based on their drink consumption, before they even start drinking.
Calculating such a predicted value is foundational in verifying the suitability of the regression model.
Residual Computation
Residuals give us critical insight into the accuracy of our predicted values from a regression model. A residual is essentially the difference between the observed value and the predicted value. In our example:
  • Observed BAC = 0.08
  • Predicted BAC = 0.041
  • Residual = 0.08 - 0.041 = 0.039
**Understanding Residuals:**
  • Residuals help us identify how much our model's prediction deviates from reality.
  • A small residual suggests a good fit for the observed data point, while a large residual indicates a poor fit.
This specific residual tells us the model underestimates the BAC for the individual after 3 drinks. Regularly computing residuals can guide us in improving our model.
Interpretation of Slope
The slope of a linear regression equation is a critical component as it reflects the relationship between the two variables being studied. In our model, the slope was given as 0.018.
  • This number tells us how much the BAC increases for each additional drink consumed.
  • Specifically, for each drink, the BAC is expected to rise by 0.018.
**Why is interpreting the slope important?**
  • It helps us understand the strength and direction of the relationship between the variables. Here, there's a positive relationship 鈥� more drinks lead to higher BAC.
  • The slope can be used to make predictions about changes in the dependent variable based on changes in the independent variable.
This interpretation is essential not just for understanding current data, but also for predicting future behavior based on past patterns.
Interpretation of Intercept
The intercept in a linear regression equation often signifies the point where the line crosses the vertical axis. In our example, the intercept was given as -0.0127.
  • This theoretically represents the predicted BAC when no drinks are consumed.
  • However, a negative BAC value does not make sense in reality.
**Understanding the Intercept:**
  • The intercept provides helpful context only when meaningful within the scenario.
  • In cases like this, the intercept should be viewed with skepticism. It shows the regression model may not perfectly represent all conditions, especially ones not grounded in reality, like zero drinks yielding a negative BAC.
The interpretation of the intercept, especially when nonsensical, reminds us to scrutinize the assumptions and constraints of our model.

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Most popular questions from this chapter

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