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

Sampling distributions for sample slopes

What is the population least-squares regression line?

• If a response variable, , has a linear relationship with an explanatory variable, , then there is a population least-squares regression line

  • This has the equation

    • where is the predicted response

    • is the population -intercept and is the population slope

What is the sample least-squares regression line?

• A random sample of observations from the population, , has sample least-squares regression line

  • This has the equation

    • where is the predicted response

    • is the sample -intercept and is the sample slope

• Different samples produce different sample least-square regression lines

  • These all have different sample slopes,

    • 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, , has

  • a mean of

    • this is the population slope

  • and a standard deviation of

    • where is the sample size

    • and where is the standard deviation of all population residuals

    • and is the standard deviation of the -values only

• In practice, is not known, so must be estimated from the sample (this estimate is called the standard error)

  • The standard error for sample slopes, , is given by the formula

    • where is the sample size

    • and where

    • and

  • Note that 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, , from the sample

    • divides by as two parameters, and , need to be estimated

What is the sampling distribution for standardized sample slopes?

• The standardized sample slope is

  • The sampling distribution for standardized sample slopes is a -distribution with degrees of freedom