Problem 1 - AP Statistics with 6 Practice Tests

Question

Inference about the slope of a least squares regression line is based on the sampling distribution of \(b\) being (A) approximately normal. (B) a chi-square distribution with \(d f=n-1\). (C) a chi-square distribution with \(d f=n-2\). (D) a \(t\) -distribution with \(d f=n-1\). (E) a \(t\) -distribution with \(d f=n-2\).

Step-by-Step Solution

Answer

E: a \(t\) -distribution with \(d f=n-2\).

Step 1: Identify the Concept

Issues related to the inference about the slope of a least squares regression line usually involve the sampling distribution of the slope coefficient.

Step 2: Determine the Distribution Type

Recall that the sampling distribution of the slope \(b\) of a least squares regression line is generally modeled by a \(t\)-distribution, not a normal or chi-square distribution.

Step 3: Degrees of Freedom

For the slope of a least squares regression line, the degrees of freedom \(d.f.\) are \(n-2\), where \(n\) is the number of data points.

Step 4: Select the Correct Answer

Given that the correct distribution is a \(t\)-distribution with degrees of freedom \(d.f.=n-2\), the answer that matches is (E).

Key Concepts

sampling distribution

When we talk about the sampling distribution, it's central to many statistical methods, including linear regression. It represents the distribution of a statistic (like the slope of a regression line) from different samples of the same size drawn from the same population. Think of it as predicting the behavior of your statistic if you were to repeatedly take samples. For the slope in least squares regression, the sampling distribution helps us understand how the slope varies across different samples. This variation tells us about the accuracy and reliability of our estimated slope.

t-distribution

The t-distribution is key in statistics when dealing with small sample sizes or unknown population standard deviations. Unlike the normal distribution which is symmetric and bell-shaped, the t-distribution is more spread out and has heavier tails. This allows it to better accommodate variations with smaller samples. In regression analysis, we use the t-distribution for inference about the slope. The reason is that the sample size might be small, and we're estimating the population parameters. When using the t-distribution, we often talk about degrees of freedom, which impacts its shape.

degrees of freedom

Degrees of freedom (df) are a concept that indicates the number of independent values or quantities which can vary in an analysis. It's crucial in the context of the t-distribution. For the slope of a least squares regression line, the degrees of freedom are calculated as `n - 2`, where `n` is the number of data points. This subtraction accounts for using two estimates from the data to calculate the slope and intercept.

least squares regression

Least squares regression is a common method for fitting a line to data points by minimizing the sum of the squares of the vertical distances (residuals) between the observed values and the values predicted by the line. This method gives us the best possible linear relationship between two variables. The slope of this regression line is particularly important as it tells us the rate of change between the variables. Understanding its distribution, inference, and accuracy is why concepts like the sampling distribution, t-distribution, and degrees of freedom are vital in regression analysis.