The regression model and its conditions

About this lesson

Statistics (AP-aligned) › Inference for regression

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The regression model and its conditions

Warm-up

Three quick questions from earlier lessons.

Discover

Two random samples of 30 students give regression slopes 2.1 and 2.6 for hours studied vs score. Why do they differ?

  1. Sampling variability: each sample estimates the same true slope
  2. One of the samples must be wrong

Key idea

A model for the population.

Population regression line: μy=β0+β1x\mu_y = \beta_0 + \beta_1 x, estimated by y^=a+bx\hat{y} = a + bx. Conditions for inference (LINER): Linear relationship, Independent observations, Normal residuals, Equal spread of residuals, Random sample or assignment.

Worked example

Checking the conditions with a residual plot.

Linear: residuals scattered around 0
No curve.
Equal spread across xx
No fan.
Roughly normal: no strong skew or outliers
Histogram of residuals.

Your turn, guided

Your turn: how many conditions are in LINER?

55
L, I, N, E, R.

Practice ★

  1. A regression on a random sample of 23 houses gives slope b. What does b estimate?
    1. The slope β₁ of the population regression line
    2. Nothing: b is the true slope
    3. The correlation r
  1. Which conditions are needed for inference about the slope from 75 pairs? Tick all that are.
    1. The true relationship is linear
    2. Observations are independent (random sample, 10% condition)
    3. Residuals are roughly normal
    4. Residuals have about equal spread for all x
    5. The correlation is above 0.5

Watch out!

Practice ★★

  1. The residual plot of a regression on 85 points fans out: residuals spread more as x grows. Which condition fails?
    1. Equal variance of the residuals
    2. Linearity
    3. Normality of the population

Exit ticket

Three questions to check you have got it.

  1. A regression on a random sample of 23 houses gives slope b. What does b estimate?
    1. The slope β₁ of the population regression line
    2. Nothing: b is the true slope
    3. The correlation r
  2. A regression on a random sample of 59 houses gives slope b. What does b estimate?
    1. The slope β₁ of the population regression line
    2. Nothing: b is the true slope
    3. The correlation r
  3. A regression on a random sample of 46 houses gives slope b. What does b estimate?
    1. The slope β₁ of the population regression line
    2. Nothing: b is the true slope
    3. The correlation r

Summary card

The regression model

y^=a+bx\hat{y} = a + bx estimates μy=β0+β1x\mu_y = \beta_0 + \beta_1 x. Conditions: LINER.

Watch out: no condition on the size of rr.

End of the lesson

Next lesson: The t statistic for the slope Back to the chapter

Answers: The regression model and its conditions

A

  1. The slope β₁ of the population regression line

B

  1. The true relationship is linear · Observations are independent (random sample, 10% condition) · Residuals are roughly normal · Residuals have about equal spread for all x

C

  1. Equal variance of the residuals

D

  1. The slope β₁ of the population regression line
  2. The slope β₁ of the population regression line
  3. The slope β₁ of the population regression line

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