Ch. 9 - Correlation and Regression

Larson - Elementary Statistics: Picturing the World 8th Edition

Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Non è quello che usi tu?Cambia libro di testo

Larson 8th Edition

Ch. 9 - Correlation and Regression

Problema 9.3.6

Capitolo 9, Problema 9.3.6

Two variables have a bivariate normal distribution. Explain what this means.

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Understand that a bivariate normal distribution describes the joint behavior of two continuous random variables, say X and Y, where their combined distribution follows a specific bell-shaped surface in two dimensions.

Recognize that each variable individually follows a normal (Gaussian) distribution, and their joint distribution is characterized by a mean vector \( \boldsymbol{\mu} = (\mu_X, \mu_Y) \) and a covariance matrix \( \Sigma = \begin{pmatrix} \sigma_X^2 & \rho \sigma_X \sigma_Y \\ \rho \sigma_X \sigma_Y & \sigma_Y^2 \end{pmatrix} \), where \( \rho \) is the correlation coefficient between X and Y.

Note that the shape of the joint distribution depends on the correlation \( \rho \); if \( \rho = 0 \), the variables are independent and the joint distribution is the product of their individual normal distributions.

Understand that the joint probability density function (pdf) of the bivariate normal distribution is given by: \[ f(x,y) = \frac{1}{2 \pi |\Sigma|^{1/2}} \exp \left( -\frac{1}{2} (\boldsymbol{z} - \boldsymbol{\mu})^T \Sigma^{-1} (\boldsymbol{z} - \boldsymbol{\mu}) \right) \] where \( \boldsymbol{z} = \begin{pmatrix} x \\ y \end{pmatrix} \).

Recognize that this distribution is useful for modeling and analyzing the relationship between two variables that are jointly normally distributed, allowing for inference about their correlation, conditional distributions, and predictions.

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Bivariate Normal Distribution

A bivariate normal distribution describes the joint behavior of two continuous random variables, where any linear combination of these variables is normally distributed. It is characterized by a mean vector and a covariance matrix, capturing the individual means, variances, and the correlation between the variables.

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Finding Z-Scores for Non-Standard Normal Variables

Joint Probability Density Function

The joint probability density function (pdf) of a bivariate normal distribution defines the likelihood of observing specific pairs of values for the two variables. It depends on the means, variances, and correlation, and its shape forms a symmetric, bell-shaped surface in two dimensions.

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Introduction to Probability

Correlation and Covariance

Correlation measures the strength and direction of the linear relationship between the two variables, while covariance quantifies how they vary together. In a bivariate normal distribution, these parameters influence the shape and orientation of the joint distribution, indicating how the variables are related.

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Correlation Coefficient

Pratica correlata

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2. Describe the explained variation about a regression line in words and in symbols."

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Domanda del libro di testo

3. What does the sample correlation coefficient r measure? Which value indicates a stronger correlation: r =0.918 or r =- 0.932? Explain your reasoning.

199

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"In Exercises 7-10, use the value of the correlation coefficient r to calculate the coefficient of determination r^2. What does this tell you about the explained variation of the data about the regression line? about the unexplained variation?

7. r =0.465"

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10. y-intercept

a. \(\hat{y}\)_i

b. y_i

c. b

d. (\(\bar{x}\), \(\bar{y}\))

e. m

f. \(\bar{y}\)"

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