Chapter 12: Linear Regression and Correlation

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are observed data points that are far from the least squares line.

observed data points that are far from the other observed data points in the horizontal direction. These points may have a big effect on the slope of the regression line.

We reject the null hypothesis. There is sufficient evidence to conclude that there is a significant linear relationship between x and y because the correlation coefficient is significantly different from zero

DO NOT REJECT the null hypothesis. There is insufficient evidence to conclude that there is a significant linear relationship between x and y because the correlation coefficient is NOT significantly different from zero.

H0→ ρ = 0

Ha→ ρ ≠ 0

The population correlation coefficient IS NOT significantly different from zero. There IS NOT a significant linear relationship (correlation) between x and y in the population.

The population correlation coefficient IS significantly DIFFERENT FROM zero. There IS A SIGNIFICANT LINEAR RELATIONSHIP (correlation) between x and y in the population.

population correlation coefficient

sample correlation coefficient

There is sufficient evidence to conclude that there is a significant linear relationship between x and y because the correlation coefficient is significantly different from zero.

"There is insufficient evidence to conclude that there is a significant linear relationship between x and y because the correlation coefficient is not significantly different from zero."

to decide whether the linear relationship in the sample data is strong enough to use to model the relationship in the population.

a number between 0 and 1 that measures how well a statistical model predicts an outcome

when expressed as a percent, represents the percent of variation in the dependent (predicted) variable y that can be explained by variation in the independent (explanatory) variable x using the regression (best-fit) line.

when expressed as a percentage, represents the percent of the variation in y that is NOT explained by variation in x using the regression line.

A positive value of r means that when x increases, y tends to increase and when x decreases, y tends to decrease.

A negative value of r means that when x increases, y tends to decrease and when x decreases, y tends to increase

is numerical and provides a measure of strength and direction of the linear association between the independent variable x and the dependent variable y.

b = r (sy / sx)

= the standard deviation of the x values.

= the standard deviation of the y values

“The slope of the best-fit line tells us how the dependent variable (y) changes for every one unit increase in the independent (x) variable, on average.”

You have a set of data whose scatter plot appears to "fit" a straight line

Helps obtain a line of best fit

estimates value of y

error or residual

measures the vertical distance between the actual value of y and the estimated value of y

the Greek letter epsilon

High values of one variable occurring with high values of the other variable or low values of one variable occurring with low values of the other variable

Looking at how close the points are to the line

shows the relationship between a dependent and independent variable(s)

uses dots to represent values for two different numeric variables.

linear regression for two variables is based on a linear equation with one independent variable.

x

y

b

a

a straight line or linear

slopes to the right

horizontal line

slopes downward to the right

two variable data

more than two variables