AP Statistics Unit 3
the least squares regression line always passes through...
(xbar, ybar)
how do you calculate residuals on the calculator?
- define L3 as the predicted values from the regression eq (Y1(L1)) - define L4 as the observed y-value - predicted y value (L2 - L3) - turn of plot 1 and the regr. eq. specify plot 2 w/ L1 as the x-var and L4 as the y-var
what are 3 limitations of correlation and regression?
- the distinction b/w explanatory and response variables is important in regression - only describe linear relationships - not resistant
how do you answer: when is a linear model appropriate?
1. you must comment on the scatterplot 2. if given or you can create a residual plot, use it 3. you may want to discuss the r-value, but it's questionable
the closer r-sq is to __________, the better the line fits the data
100%
you can describe the overall pattern of a scatterplot by the
FODS (form, outliers, direction, strength)
what is the fraction of the variation in the values of y that is accounted for by the least squares regression line of y on x?
coefficient of determination
how do we interpret two variable graphs?
fods
what is the slope template?
for every 1 point increase in (x-variable) there is a corresponding (number) (increase/decrease) in (y-variable).
sse is always ________ than sst
less
does a correlation close to 1 or -1 always indicate a correlation?
no, it's only for a linear relationship
is correlation resistant to extreme observations?
no. r is strongly affected.
residual =
observed - predicted
what is a least squares regression line?
of y on x is is the line that makes the sum of the squared residuals as small as possible
what's the difference between positive and negative association?
positive - above avg values of one tend to lead to above avg values of the other (same w/ below avg) negative - above avg value of one tend to accompany below avg of another
what is the coefficient of determination?
r-sq
what is the typical prediction error represented by?
r-sq
what is the r-sq template
r-sq% of the variation in y-var can be explained by the LSR on x-var
what is the equation to find slope?
slope = r (Sy/Sx)
what does the standard deviation of residuals tell us?
the approximate size of a typical/average prediction error (residual)
what is a residual?
the difference between an observed value of the response variable and the value predicted by the regression line
what does correlation measure?
the direction and strength of the linear relationship b/w two quantitative variables
which variable is on the horizontal axis?
the explanatory variable (if there is one)
how are the response and explanatory variables related to dependent and independent variables?
the explanatory variable is the independent variable, and the response variable is the dependent variable
what does increasing/decreasing spread about the residual line indicate?
the line is better for smaller (or larger) x-values
what special property do residuals have?
the mean of the least squares residuals is always 0, because you've standardized and the mean of standardized data is 0
what is the difference between a response variable and explanatory variable?
the response variable measures the outcome and the explanatory variable may help explain or influence changes in a response variable
what is extrapolation and why is it dangerous?
the use of a regression line for prediction far outside the interval of values of the explanatory variable x used to obtain the line. these predictions are usually not accurate.
what does it mean if two variables have high correlation?
their R is close to 1 or -1. as one increases, so does the other and as one decreases so does the other
what does it mean if two variables have no correlation?
their r is 0. the variables do not have any association
what does it mean if two variables have weak correlation?
their r is close to 0. there is barely a correlation.
when are scatterplots appropriate?
they are the most useful graph for displaying 2 quantitative variables
if r^2 = .95, what can be concluded about the correlation between x and y?
they have a very strong correlation
If a least-squares regression line fits the data well, what characteristics should the residual plot exhibit?
uniform scatter. residuals should be relatively small in size
when is an outlier an influential observation?
when removing it would markedly change the result of the calculation
why must two variables be quantitative in order to find the correlation between them?
you can't find correlation with a categorical variable because it has no numbers and the formula requires numbers
the mean of the least squares residuals is always...
zero
the sum of the residuals equals...
zero
why does association not imply causation?
because there could be a lurking variable
what is true about the correlation is the r-value is... a. near 0 b. near 1 c. near -1 d. exactly 1 e. exactly -1
a. very weak b. very strong positive c. very strong negative d. perfect positive linear relationship e. perfect negative linear relationship
what is the formula for the least squares regression line?
b = Sy/Sx a = (ybar) - b(xbar) xbar = x mean ybar = y mean a = y-intercept b = slope Sy = standard deviation of y values Sx = standard deviation of x values
what is a regression line?
a line that describes how a response variable y changes as an explanatory variable x changes
what is a residual plot?
a scatterplot of the residuals against the explanatory variable
what is a lurking variable?
a variable that influences both x and y even though there is no direct correlation b/w x and y