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Chapter 7: Problem 25

The equation of the least-squares regression line for predicting points earned from spending is $$ \text { points }=39.64+0.77 \times \text { spending } $$ We use the regression equation to predict the points a team would earn if they spent no money. We conclude that a. the team will earn \(39.64\) points. b. the prediction is not sensible because the prediction is far outside the range of values of the response variable. c. the prediction is not sensible because no money is far outside the range of values of the explanatory variable. d. the prediction is not sensible because of the outlier present in the data.

Short Answer

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c. The prediction is not sensible because no money is far outside the range of values of the explanatory variable.

Step by step solution

01

Understanding the Equation

The regression equation given is \( \text{points} = 39.64 + 0.77 \times \text{spending} \). This indicates that the expected points increase by 0.77 for every additional unit spent.
02

Setting Spending to Zero

We are asked to predict the points when the spending is zero. To do this, we substitute \( \text{spending} = 0 \) into the equation: \( \text{points} = 39.64 + 0.77 \times 0 \).
03

Calculating the Prediction

Substituting \( \text{spending} = 0 \) into the equation simplifies to \( \text{points} = 39.64 + 0 \), which results in \( \text{points} = 39.64 \). Thus, if the team spends no money, they are predicted to earn 39.64 points.
04

Interpreting the Options

Option (a) states that the team will earn 39.64 points. While this calculation is correct mathematically, option (c) should be considered, as spending no money may fall outside the range of past data used to form this regression.
05

Selecting the Correct Option

Considering the realistic scenario, option (c) is appropriate: the prediction might not be sensible if 'no money' is beyond the data's explanatory range, impacting the prediction model's reliability.

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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 is a mathematical representation that models the relationship between two variables. In our context, it is used to predict the number of points earned based on the amount spent, formulated as:\[\text{points} = 39.64 + 0.77 \times \text{spending}\]This equation tells us two key things:
  • 39.64 is the intercept, which is the predicted points when the spending is zero.
  • 0.77 is the slope, indicating an increase of 0.77 points for each additional unit of spending.
Regression equations help us understand trends and make predictions. They can be useful in various fields, such as economics, biology, or marketing, to model how changes in one variable affect another.
Explanatory Variable
The explanatory variable is the independent variable in a regression model. It is the variable that is manipulated to observe the effect on the response variable. In this case, 'spending' is the explanatory variable.
  • It's important because it provides the input to the regression equation.
  • Helps to explain or predict changes in the response variable (points earned).
When using a regression model, the range of values the explanatory variable takes is critical. If the current data range doesn't cover the zero spending scenario, predictions made in such cases might not be valid or reliable. Understanding the role of the explanatory variable is crucial for making sensible predictions.
Response Variable
The response variable is the dependent variable, whose variation we are trying to explain with the model. In our exercise, 'points earned' is the response variable. This variable is:
  • What we are predicting or explaining using the regression model.
  • Affected by changes in the explanatory variable.
For accurate predictions, the response variable should cover a wide range of scenarios. Misestimation might occur if points earned in reality don't align well with our model's predictions due to limitations in the data range or other dataset anomalies such as outliers.
Range of Data
The range of data refers to the scope within which the explanatory variable's observed values lie. It is a critical aspect in regression analysis.
  • Ensures that the regression model remains valid by staying within the parameters used to create it.
  • Predicting outside this range can lead to unreliable conclusions, known as extrapolation.
In our example, predicting points for zero spending must be carefully considered if zero is outside the observed spending range. This is because the model was likely not created with zero spending in mind. Understanding the range helps determine the feasibility and reliability of predictions, ensuring credible and accurate usage of the regression model.

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