Statistics: Chapter 7

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Interpret R^2

% of the variability in remaining (y-context) can be explained by its linear relationship with (x-context)

Problems with residual plots?

-Curved pattern -Outliers -Fanning

What is a negative residual?

A negative residual tells us OBSERVED is less than the PREDICTED (OVERESTIMATED)

What is a positive residual?

A positive residual tells us OBSERVED is more than predicted (UNDERESTIMATED)

How should a scatterplot of the residuals look like?

A scatterplot of the residuals should be the most boring scatterplot you've ever seen; it shouldn't have any interesting features like direction or shape; it should have the same amount of scatter throughout without bends or outliers

A line that fits WELL has...

A very small residual

After we fit a regression model, we usually plot the...

After we fit a regression model, we usually plot the residuals in the hope of finding NOTHING

What is an R^2 of 100%?

An R^2 of 100% is a perfect fit with no scatter around the line; the Se would be zero

What is the y-intercept?

B0=(mean of the y)-(B1)(mean of x)

What is the slope?

B1= rSy/Sx

Interpret the slope

Based on the model, for each additional (x-context), we expect a (increase or decrease) in (y-context) of # on average

Interpret the y-intercept

Based on the model, when (x-context) is zero, the expected (y-context)!is # on average

Will changing the units of the variables affect the correlation?

Changing the units of the variables doesn't change the correlation

What does each predicted y tend to do?

Each predicted y tends to be closer to its mean (in standard deviations) than its corresponding x was

Residual plot

Ideally: NO PATTERN

r= 0.8-0.5

MODERATE

What happens when you move any number of standard deviations in x?

Moving many number of standard deviations in x, moves r times that number of standard deviations in y

What is r=0?

NO LINEAR RELATIONSHIP

What is r=1 or r=-1?

PERFECT linear association

What is R^2?

R^2 is the percentage of variability in y that is explained by x

r= 1.00-0.8

STRONG

Interpret the Standard Deviation of Residuals

The AVERAGE distance between the actual (observed) y-context values and the predicted y-context values is 's' units

What does the size of the residuals tell us?

The SIZE of the residuals tells us how well the line fits over the data

The smaller the sum...

The better the fit

The closer the R^2 is to 100% then..

The less scatter there is around the line

What is the line of best fit?

The line of best fit is the line for which the sum of the squared residuals is smallest (the least squares line)

What is residual?

The residual is the difference between the observed value and its associated predicted value residual= observed-predicted

What does the residual value tell us?

The residual value tells us how far off the model's prediction is at that point

Should the residuals all share the same underlying spread?

The residuals should all share the same underlying spread, so we make sure that the residual plot has about the same amount of scatter throughout

What does the standard deviation of the residuals (Se) gives us?

The standard deviation of the residuals gives us a measure of how far the points spread around the regression line

What does the variation in residuals (R^2) give us?

The variation in residuals is the key to assessing how well the model fits

How do you find the residual?

To find the residuals, you always subtract the: observed-predicted

Does units matter for the slope?

Units do matter for the slope

r= 0.5-0

WEAK

What do we call the estimates made from a model?

We call the estimates made from a model, the predicted value and write it as y-hat

Is it a good idea to make a histogram of the residuals?

Yes because if we see a UNIMODAL, SYMMETRIC histogram then we can apply the 68-95-99.7 rule to see how well the regression model describes the data

What variations do you compare?

You compare the variation of the response variable with the variation of the residuals

What is the regression line?

y-hat=B0+B1(x)


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