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Understanding Father-Son Heights Relationship in Simple Linear Regression Model

Explore the relationship between father's and son's heights using Galton's survey data in a simple linear regression model. Learn about assumptions, inferences, and properties of estimators in Utopia JMP simulations.

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Understanding Father-Son Heights Relationship in Simple Linear Regression Model

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  1. Stat 112: Notes 2 • This class: Start Section 3.3. • Thursday’s class: Finish Section 3.3. • I will e-mail and post on the web site the first homework tonight. It will be due next Thursday.

  2. Father and Son’s Heights • Francis Galton was interested in the relationship between • Y=son’s height • X=father’s height • Galton surveyed 952 father-son pairs in 19th Century England. • Data is in Galton.JMP

  3. Simple Linear Regression Model

  4. Sample vs. Population • We can view the data – -- as a sample from a population. • Our goal is to learn about the relationship between X and Y in the population: • We don’t care about how father’s heights and son’s heights are related in the particular 952 men sampled but among all fathers and sons. • From Notes 1, we don’t care about the relationship between tracks counted and the density of deer for the particular sample, but the relationship among the population of all tracks; this enables to predict in the future the density of deer from the number of tracks counted.

  5. Simple Linear Regression Model

  6. Checking the Assumptions

  7. Residual Plot

  8. Checking Linearity Assumption

  9. Violation of Linearity

  10. Checking Constant Variance

  11. Checking Normality

  12. Inferences

  13. Sampling Distribution of b0,b1 • Utopia.JMP contains simulations of pairs and from a simple linear regression model with • Notice the difference in the estimated coefficients from the y’s and y*’s. • The sampling distribution of describes the probability distribution of the estimates over repeated samples from the simple linear regression model with fixed.

  14. Utopia Linear Fit y = 1.4977506 + 0.9876713 x Parameter Estimates Term Estimate Std Error t Ratio Prob>|t| Intercept 1.4977506 0.300146 4.99 <.0001 x 0.9876713 0.016907 58.42 <.0001 Linear Fit y* = 0.9469452 + 1.0216591 x Parameter Estimates Term Estimate Std Error t Ratio Prob>|t| Intercept 0.9469452 0.364246 2.60 0.0147 x 1.0216591 0.020517 49.79 <.0001

  15. Sampling distributions • Sampling distribution of • Sampling distribution is normally distributed. • Sampling distribution of • Sampling distribution is normally distributed. • Even if the normality assumption fails and the errors e are not normal, the sampling distributions of are still approximately normal if n>30.

  16. Properties of and as estimators of and • Unbiased Estimators: Mean of the sampling distribution is equal to the population parameter being estimated. • Consistent Estimators: As the sample size n increases, the probability that the estimator will become as close as you specify to the true parameter converges to 1. • Minimum Variance Estimator: The variance of the estimator is smaller than the variance of any other linear unbiased estimator of , say

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