Unit 3 - Relationships (RBA)

Explanatory Variable

x-axis, independent

Response Variable

y-axis, dependent

Scatter Plot Title

Y vs. X

What do scatterplots show?

the relationship between 2 quantitative variables

How do you describe scatterplots?

direction, form, strength, outliers

Direction

Positive/Negative

Form

Linear vs. Nonlinear

Strength

Strong/Weak OR Moderately Strong/Moderately Weak

Outlier

can include from appearance -- clusters?

Which value measures correlation?

r value -- measures direction & strength

What numbers is the r value ALWAYS between?

-1 and 1

What does r > 0 mean?

positive association

What does r < 0 mean?

negative association

What do r-values near 0 indicate?

a weak linear relationship

What does the least squares regression line summarize?

the relationship between two variables

What is the LSRL?

the line closest to all points

What does the LSRL describe?

how y changes as x changes -- can predict y for value x

What is the model to interpret a regression line?

ŷ = a + bx

y-hat (ŷ)

predicted value of the response variable y

b-value (LSRL)

the slope: the amount by which y is predicted to change when x + 1

a-value (LSRL)

the y-intercept: the predicted value of y when x = 0

How do you write the interpretation of the slope?

As the (explanatory variable) increases by one (units), the (response variable) is predicted to change by _____(units).

How do you write the interpretation of the y-intercept?

When the (explanatory variable) is 0 (units), it is predicted that there will be _____(units) of (response variable).

Extrapolation

using regression lines for predictions far outside the interval - causes inaccurate and skewed data

Residuals

the difference between an observed value and the value predicted by the regression line

Residual equation

residual = observed y - predicted y (y - ŷ)

What question does finding the residual answer?

How far is the data point from the LSRL?

What question will plotting residuals answer?

Is a linear model appropriate for the data?

How does the residual plot show if a linear model is appropriate?

it should show no obvious patterns and the residuals should be relatively small in size

Standard deviation of residuals

on average, how far away the data points are from the regression line

s-value

standard deviation of residuals - gives approximate size of a "typical" prediction error

How do you write the interpretation for the s-value?

if s = (value), the predictions using the least squares regression line will be off by about (s-value) -- above or below

What is the r-squared value?

coefficient of determination

How do you write the interpretation for the r-squared value?

__% of the variation in [response variable] is accounted for by the linear model relating [response variable name] to [explanatory variable name]

What is described by the r-squared value?

how much of the variation is explained by the independent variable

What is the reasoning behind the name of the LSRL?

it is the line that makes the sum of the squared residuals as small as possible.