Question
What is the difference between the following two regression equations? \[ \hat{y}=b_{0}+b_{1} x \] Choose the correct answer below. A. The first equation is for sample data; the second equation is for a population. \[ \mathrm{B} \text {. The first equation is for a population; the second equation is for sample data. } \]
Ask by Wood Ford. in the United States
Nov 23,2024
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Answer
A. The first equation is for sample data; the second equation is for a population.
Solution
The correct answer is:
A. The first equation is for sample data; the second equation is for a population.
In regression analysis, the equation \(\hat{y} = b_0 + b_1 x\) typically represents the estimated relationship based on sample data, where \(b_0\) and \(b_1\) are the estimated coefficients derived from the sample. In contrast, the population regression equation would generally be represented as \(y = \beta_0 + \beta_1 x\), where \(\beta_0\) and \(\beta_1\) are the true population parameters.
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The difference between the two regression equations lies in what they represent. The equation \( \hat{y} = b_0 + b_1 x \) typically represents the fitted line for sample data, where \( b_0 \) is the intercept and \( b_1 \) is the slope derived from a sample. This equation is an estimate of the relationship within the sample population. On the other hand, when discussing a full population, one would use the true parameters \( \beta_0 \) and \( \beta_1 \) instead of the sample estimates, frequently denoted as \( Y = \beta_0 + \beta_1 X \). The transition from sample to population indicates a shift from estimation to actual parameters, reflecting a broader context.
