Unit 7

Question 1

Why do we use scatterplots?

Answer 1

To graph the relationship between two quantitative variables.
When we have two variables that we believe are related to one another. With two variables, we cannot use a histogram.

Question 2

What is a scatterplot?

Answer 2

• each point on a scatter plot shows the value of two different variables measured on the same individual
• if you have an explanatory variable and a response variable they should be plotted on the x-axis and y-axis respectively
• if there is no explanatory variable, the labelling of the axis is unimportant

Question 3

What are the two different types of models to model a variable y by variables x

Answer 3

1. deterministic model: a model where y can always be predicted exactly by x alone
2. probabilistic model: a model that includes both a deterministic portion and a random error term

Question 4

What is the general form of a probabilistic linear model?

Answer 4

y-intercept + slope*x + random error term (which follows a normal distribution of N(0, sigma)

Question 5

How does the scatterplot affect the sigma value used for the random error term?

Answer 5

points spread out -> sigma large
points close to the line -> sigma small

Question 6

What does the probabilistic linear model look like for average y?

Answer 6

mean-y = y-intercept + slope*x (the errors should average out to 0 since the error follows N(0, sigma))

Question 7

What is the regression line?

Answer 7

The line we will use to model our data and make predictions.
the predicted value of y for a given x = the estimate of the intercept + the estimate of the slope*x

Question 8

What is the difference between observed and predicted values of y?

Answer 8

Observed values of y are the points on our scatterplot. Our predicted values of y are the values along the line we fit.

Question 9

What is the relationship between the observed and predicted values of y called?

Answer 9

The vertical difference between them is the “error”; ow far off our real world value is from our modeled/predicted value

Question 10

What do we want to minimize in fitting our curve?

Answer 10

the sum of (observed y - predicted y)^2. Not the sum of the differences because positives and negatives would cancel each other out like in variance

Question 11

What are the values of the estimate of the intercept and the estimate of the slope that minimize the squared errors?

Answer 11

estimate of slope: SSxy / SSxx
estimate of intercept: sample y - estimate of slope*sample x

Question 12

What does the slope of the regression line (beta-1-hat) tell us?

Answer 12

How much y changes when x changes by one

Question 13

What is the general form for an interpretation of beta1-hat?

Answer 13

For every unit increase in X we predict that Y will increase(decrease) by beta-1-hat units

Question 14

What is the general form for an interpretation for beta0-hat?

Answer 14

The predicted value of Y when X is zero (if x = 0 is outside the range of the data, this interpretation may be meaningless or nonsensical)

Question 15

What are the assumption we need to make for our regression to be valid?

Answer 15

1. the expected value (the mean) of the errors is 0
2. the variance of the error terms in constant across all values of y
3. the data set is linear
4. the probability distribution of the errors is normal
5. the errors are independent between observations

Question 16

What is a residual plot?

Answer 16

A way to test if the variance of the error terms is constant and if the dataset is linear. We plot the predicted values against the errors. If there doesn’t appear any section that’s obviously much taller than any other, the variance of the errors is constant. If there is no discernable pattern in the plot, the data is linear.

Question 17

How can we test if the probability distribution of the errors is normal?

Answer 17

Make a normal quantile plot of the error terms. In particular, we make a normal quantile plot of the standardized residuals

Question 18

What are standardized residuals?

Answer 18

Error terms converted to z-scores

Question 19

What does the error variance tell us about our model?

Answer 19

The bigger the variance, the worse our prediction will be. The more the errors very from 0, the more error there is in the model

Question 20

How can we find an estimate for the variance of the error?

Answer 20

sigma-hat = sqrt(SSyy - beta1-hat*SSxy / (n - 2) )

Question 21

What is the general interpretation of standard error?

Answer 21

Approximately 95% of our observations will fall within two standard errors of the regression line

Question 22

What would our model look like if x had no relationship with y?

Answer 22

y would equal the population mean of y plus or minus some error (beta-1-hat = 0)

Question 23

What is the correlation of a sample?

Answer 23

r (a number between -1 and 1) measures the strength of a linear relationship between two variables. It can tell us both the strength and the direction (positive or negative) of the relationship

Question 24

What are the properties of correlation?

Answer 24

1. positive r values indicate a positive association. Negative r values indicate a negative association
2. r falls between -1 and 1 inclusively
3. r values near -1 and 1 indicate a strong linear relationship. r values near 0 indicate a weak linear relationship.
   4.. If r = -1 or r = 1, the points fall exactly on a straight line and are perfectly correlated

Question 25

What is a lurking variable?

Answer 25

A variable that helps explain the relationship between variables in a study but which is not itself included in the study

Question 26

What is the coefficient of determination?

Answer 26

r squared: it helps us establish how much of y’s movement is explained by its relationship to x and how much is due to error/other factors. Of all the movement in our y, what proportion of it can be attributed to a relationship with x. The better the points fit to the line, the higher this value will be

Question 27

What does a coefficient of determination of zero mean?

Answer 27

x explains nothing about the variation in y and it’s all due to error

Question 28

What does a coefficient of determination of one mean?

Answer 28

x explains all of the variation in y and there is no error

Question 29

What are the properties of r squared?

Answer 29

1. r squared takes on values between 0 and 1 inclusive
2. if r squared = 1 we can predict Y exactly from X and we say the regression accounts for all the variability of Y
3. if r squared = 0, regression on X tells us nothing about the value of Y

Question 30

What is the residual?

Answer 30

observed - predicted. The error of our estimate.

Question 31

What does least-squares regression actually do?

Answer 31

If we were to just minimize the errors, positive and negative errors would cancel out. We square instead of take the absolute value to put a penalty on extreme errors.

Question 32

What are outliers in a two-variable model?

Answer 32

Points that either do not fit in with the general pattern and/or range of our other x and y values. They could effect the slope of our regression line because it is based on the sums of squares which can be huge if we have points very far away from the mean.

Question 33

What are the different kinds of outliers in a two-variable model?

Answer 33

• x direction (outside of the general range of our x points. Creates massive error term)
• y direction (the slope doesn’t drastically change. If we tilt the line too much to meet it, we would create really bad outliers on the other side)
• x and y direction (may not change the slope, but will change average x and y and hence sums of squares)
• outside of general pattern (violates normality of errors assumption)