AP®StatisticsCollege BoardRevision NotesUnits 6-9: InferenceInference for Regression SlopesSampling Distributions for Sample Slopes

Sampling Distributions for Sample Slopes (College Board AP® Statistics)

Revision Note

Mark Curtis

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Expertise

Maths

Sampling distributions for sample slopes

What is the population least-squares regression line?

If a response variable, $y$ , has a linear relationship with an explanatory variable, $x$ , then there is a population least-squares regression line
- This has the equation $\hat{y} = α + β x$
  - where $\hat{y}$ is the predicted response
  - $α$ is the population $\hat{y}$ -intercept and $β$ is the population slope

What is the sample least-squares regression line?

A random sample of $n$ observations from the population, $(x_{i}, y_{i})$ , has sample least-squares regression line
- This has the equation $\hat{y} = a + b x$
  - where $\hat{y}$ is the predicted response
  - $a$ is the sample $\hat{y}$ -intercept and $b$ is the sample slope
Different samples produce different sample least-square regression lines
- These all have different sample slopes, $b$
  - This means sample slopes have a sampling distribution

What is the mean and standard deviation of the sampling distribution for sample slopes?

The sampling distribution for sample slopes, $b$ , has
- a mean of $β$
  - this is the population slope
- and a standard deviation of $\frac{σ}{σ_{x} \sqrt{n}}$
  - where $n$ is the sample size
  - and where $σ$ is the standard deviation of all population residuals
  - and $σ_{x} = \sqrt{\frac{\sum_{}^{} {(x_{i} - \bar{x})}^{2}}{n}}$ is the standard deviation of the $x$ -values only
In practice, $\frac{σ}{σ_{x} \sqrt{n}}$ is not known, so must be estimated from the sample (this estimate is called the standard error)
- The standard error for sample slopes, $s_{b}$ , is given by the formula $s_{b} = \frac{s}{s_{x} \sqrt{n - 1}}$
  - where $n$ is the sample size
  - and where $s = \sqrt{\frac{\sum_{}^{} {(y_{i} - {\hat{y}}_{i})}^{2}}{n - 2}}$
  - and $s_{x} = \sqrt{\frac{\sum_{}^{} {(x_{i} - \bar{x})}^{2}}{n - 1}}$
- Note that $s$ is an estimate for the standard deviation of all population residuals (the standard error of the residuals) and its formula:
  - is based on the residuals, $y_{i} - {\hat{y}}_{i}$ , from the sample
  - divides by $n - 2$ as two parameters, $α$ and $β$ , need to be estimated

Exam Tip

All the formulas required for the sampling distributions of sample slopes are given in the exam.

What is the sampling distribution for standardized sample slopes?

The standardized sample slope is $\frac{b - β}{s_{b}}$
- The sampling distribution for standardized sample slopes is a $t$ -distribution with $n - 2$ degrees of freedom

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Mark Curtis

Author: Mark Curtis

Mark graduated twice from the University of Oxford: once in 2009 with a First in Mathematics, then again in 2013 with a PhD (DPhil) in Mathematics. He has had nine successful years as a secondary school teacher, specialising in A-Level Further Maths and running extension classes for Oxbridge Maths applicants. Alongside his teaching, he has written five internal textbooks, introduced new spiralling school curriculums and trained other Maths teachers through outreach programmes.