Chapter 3: Scatterplots & Correlation

How do we describe bivariate (two-variable explanatory-response variable) when graphed?

FUDS:

• Form

  • Linear vs. non-linear

• Unusual features

  • Outliers

  • Gaps

• Direction

  • Positive association vs. negative assoiciation (slope)

• Strength

  • Strong vs. weak correlation

what happens to strength and slope when you switch x and y

strength stays the same; slopes changes

Does changing units (like miles → km) change strength?

no

Correlation coefficient / interpret the value of r

correlation coefficient = r value

• measures strength of association (closer to 1 = stronger)

• ex: if r = 0.86, say there is a strong, positive relationship btwn x and y

Least Squares Regression Line equation

LSRL: ŷ = a + bx

• ŷ = predicted y value

• a = y-int

• b = slope

Always define ŷ and x; either by saying “where ŷ = ___ and x = ___” or put the actual context in place of ŷ and x and put a hat over the ŷ

To get the LSRL, do stat → calc → #8 (LinReg (a + bx))

Residual equation

Residual= y - ŷ

• Residual is the difference btwn an observed value of the response variable and the value predicted by the regression line

• If the residual is positive, then the actual value lies ABOVE the predicted value

• If the residual is negative, then the actual value lies BELOW the predicted value

Interpret the slope in context

For every increase of one unit in the x, the predicted y will increase/decrease by about slope according to the LSRL

ex: for every increase of one degrees temp, the predicted number of ice creams sold will increase by about 4 according to the LSRL

Interpret the y-intercept in context

When x = 0, the predicted y is y-int according to the LSRL

ex: when it is 0 degrees, the predicted number of ice creams sold is 10 according to the LSRL

Coefficient of determination / interpretation

• Coefficient of determination = r²

• (r²)% of the variation in y can be explained by x according to the LSRL

• ex: if r² = 0.70 → 70% of the variation in ice creams sold can be explained by temperature according to the LSRL

Get r from r²

• Square root r² as a decimal

• If slope is positive, then r is positive

• If slope is negative, then r is negative

How to get a residual plot

• Put x data in L₁

• Put y data in L₂

• Do linreg(ax + b) to store the data in your calculator

• Go to L₃ → list → RESID

• Stat plot → x-list: L₁, y-list: L₃

Interpret the residual plot

If the data in the residual plot is random and scattered, then a linear model is the best fit for the data

Extrapolation

be careful predicted values outside the data range; the linear relationship may not continue

How to read a minitab output