chapter 5: regression

Question 1

𝑦̂ = 𝑎 + 𝑏x

Answer 1

This is how you determine the least squares regression line, where “y-hat” gives a predicted response for any x.

Question 2

slope b (different from the other b) in the least squares line is calculated as:

Answer 2

The slope of the regression line is the product of the correlation and the standard deviation of y over the standard deviation of x. r times (the standard deviation of y divided by the standard deviation of x)

Question 3

𝑎 = 𝑦̅ − 𝑏x̄

Answer 3

for determining the least squares resgression line, where a is the mean of y minus the slope times the mean of x

Question 4

Truse or false: we should always give 𝑟² along with our line to show how valid it is

Answer 4

True, because a least squares regression line can always be created regardless

Question 5

To extrapolate

Answer 5

is to use values predicted by the line outside of the range of our 𝑥-values, which we should avoid

Question 6

a residual

Answer 6

is the vertical distance between any given datapoint and the least-squares regression line

Question 7

influential points

Answer 7

observations that have values that have more effect on the calculations than the rest, or that would drastically change the calculations if they were removed… If the discrepancy is only in the 𝑥 direction (or in the explanatory variable), then the influential point may affect the regression line; if it is in either the 𝑥 or 𝑦 direction (or in either variable), then the influential point may affect the correlation and/or the slope of the line

Question 8

criteria to determine whether or not it‘s likely that there is a causal relationship

Answer 8

• The association is strong.
• The association is consistent across different datasets
• Higher values of explanatory variable are associated with higher values of response variable.
• The explanatory variable precedes the response variable (in time).
• The explanatory variable is plausible as a cause of the response variable.

Question 9

r²

Answer 9

gives the percentage of variation in y that is explained by the least squares regression line.

Question 10

ecological correlation

Answer 10

correlation based on averages (not to be trusted)

Question 11

lurking variables

Answer 11

Lurking variables are always potential problems in observational studies. Experiments are necessary to exclude the effect of lurking variables so that we can draw conclusions about the explanatory variable causing changes in the response variable.

Question 12

residual formula

Answer 12

observed y - predicted y

Question 13

the four statistics related to regression

Answer 13

slope, y-intercept, correlation, and r²

Question 14

Interchanging x and y always changes:

Answer 14

slope and y-intercept, but not correlation

Question 15

to calculate correlation…

Answer 15

take the root of r² aka the percentage of variation in y that x explains