Statistics 6

Question 1

Outline 3 characteristics of explanatory statistics

Answer 1

1. Most powerful form of statistical analysis
2. Determines causation of relationship
3. Strick assumptions

Question 2

What does regression allow us to do with statistics?

Answer 2

It allows us to make a numerical prediction of how one variable linearly affects another

Question 3

What does carrying out regression allows us to define?

Answer 3

Causation relationship - which variables are independent and which are dependent i.e. which one affects the other and by how much

Question 4

What is the regression line?

Answer 4

Numerical description of the line of best fit

Question 5

What are the three assumptions/conditions of a regression line?

Answer 5

Continuous data
Parametric (Normally distbd. and n>30)
There will always be scatter from the perfect relationship

Question 6

What is the standard format for expressing a regression line? What are the components of this format?

Answer 6

Y = a + bx [a = y-intercept and b = regression coefficient]

Question 7

What criteria is the regression line based of? explain this criteria

Answer 7

“least-squares criterion” - ensuring that there is equal total distance of points from the line either side of the line (total distance of points above line = total distance below line). This can be satisfied in many lines and so we need to make sure that the line drawn is the best fit between all the points

Question 8

What two characteristics do we need to look at to determine whether the regression line is useful?

Answer 8

Unexplained variance and explained variance

Question 9

What is explained variance?

Answer 9

The variation of the points from the line that can be understood/ explained by the regression line. The closer the points are to the line (in between the regression line and the mean y-value) the higher the explained variances

Question 10

What is unexplained variance?

Answer 10

The variation of the points from the line that are not well understood/explained by the regression line that has been drawn. The further away the points are from the regression line (not on the side with the mean y-value, the higher the unexplained variance)

Question 11

How do you define a useful regression line and a bad regression line based off the explained and unexplained variance?

Answer 11

The higher the explained variance and the lower the unexplained variance, the better the regression line is because it represents a greater amount of the dataset. If this is reversed then the regression line is not very representative as it does not explain much of the data.

Question 12

What is another word for the unexplained variance?

Answer 12

Residuals

Question 13

If you are visually determining the usefulness of the regression line, what do you need to do and what is another key component?

Answer 13

You need to compare the height of the unexplained variance and the explained variance. To do this, measure the height difference of the regression line to the point in question and then compare it against the height difference from the regression line from the same point to the mean y-value.

Question 14

What is the f-ratio?

Answer 14

Value which represents the ratio of the explained variance to the unexplained variance.

Question 15

What is the notation for the explained variance?

Answer 15

S (subscript y and superscript 2)

Question 16

What is the notation for the unexplained variance?

Answer 16

S (subscript e and superscript 2)

Question 17

What is the best way to remember the different notations for the explained and unexplained variance?

Answer 17

the unexplained variance notation has an e which you would think should represent the explained variance - but no. Instead, they are the other way round so just think of them as opposites to what they represent.

Question 18

How do you calculate the f-ratio and what does the result mean?

Answer 18

Explained variance divided by unexplained variance. If >1 then this means the explained variance is bigger than the unexplained variance, if <1 then this means the unexplained variance is bigger. We can then use this to determine the usefulness of the regression line. the bigger the value of f the better the regression line.

Question 19

What is the coefficient of explanation?

Answer 19

It determines what proportion of the variance in Y can be explained by the variance in X

Question 20

What notation is used for the coefficient of explanation?

Answer 20

r superscript 2

Question 21

What would a coefficient of explanation that is 0.85 mean?

Answer 21

that 85% of the variance in y can be explained by the variance in X

Question 22

What can the coefficient of explanation range between and what does the sign of this value mean? What calculation is it similar to?

Answer 22

-1 to +1: if positive or negative then this indicates the direction of the relationship. Similar to PMCC

Question 23

Using the knowledge of what the coefficient of explanation value means, what does this mean about the explained and unexplained variance?

Answer 23

If the value is high then this means that the explained variance is also high

Question 24

What are the two source of error in regression models?

Answer 24

1. Standard error of residuals = larger scatter around the regression line i.e. the error is that the residuals are very scattered
2. Sampling error = regression line characteristics error (i.e. y-intercept and coefficient gradient)

Question 25

What is uncertainty/error in the y-intercept of the regression line represented as?

Answer 25

lines either side of and equidistance to the drawn regression line to indicate that the error could go either way

Question 26

What is uncertainty/error in the regression coefficient of the regression line represented as?

Answer 26

Lines travelling at opposite angles to the drawn regression line through the same point. They are angled alternately to the line.

Question 27

What is the combined uncertainty/error in the y-intercept and regression coefficient (total sampling error) represented as?

Answer 27

Two lines that curve toward and then away from the regression line either side of it

Question 28

Is the sampling error normally represented on a plot with the regression line?

Answer 28

Yes

Question 29

What is the purpose of the standardized residual calculation?

Answer 29

To identify any usual residuals

Question 30

What is homoscedacity?

Answer 30

Residuals are consistent across the x-axis i.e. there is no real change in their variation across the distribution when you are predicting y form x which is really important

Question 31

What is autocorrelation?

Answer 31

Where each correlation between x and y is not independent of each other

Question 32

What is the problem of autocorrelation?

Answer 32

Regression assumes that autocorrelation does not happen i.e. each correlation result is independent of each other

Question 33

What is the Durbin-Watson statistic?

Answer 33

A numerical statistic for determining autocorrelation

Question 34

What is the coefficient of explanation statistic represented by in SPSS?

Answer 34

“R. square value”

Question 35

What is the t-value in SPSS?

Answer 35

The statistical significance of the regression coefficient value

Question 36

How do you interpret the Durbin-Watson statistics?

Answer 36

=2 means no autocorrelation
<2 means positive autocorrelation
>2 means negative autocorrelation