STAT MOD 3: Chapter 4

Question 1

What is a scatter plot? What are the two variables?

Answer 1

A point represents combination of two measurements for an individual observation
- for bivariate, numeric
- Explanatory variable (x axis) and response/dependent variable (y axis)

Question 2

What is the form of a relationship? What are the two types of relationship?

Answer 2

Average pattern or form of the scatter plot

Linear: pattern of relationship resembles a straight line
Curved: pattern of relationship resembles a curve

Question 3

What is direction? What are the two types of direction?

Answer 3

IF linear, you can determine direction of relationship

Positive association: when values of one variable increases as value of the variable increases (move in same direction)

Negative association: when values of one variable decrease as value of the other variable increase (move in opposite directions)

Question 4

What is strength of a relationship?

Answer 4

when our points follow a pattern (linear or curved) without a lot of scatter

Question 5

What is correlation coefficient (r)?

Answer 5

numerical objective measure that indicates
- strength
- direction
of linear relationship between two numeric variables

can range between -1 and 1

Question 6

How do you interpret correlation coefficient?

Answer 6

Sign of (r) indicates the strength and direction of relationship
- can range between -1 and 1

Between -0.5 and 0.5 = weak
Between -0.8 and -0.5/0.5 and 0.8 = moderate
Between -1 and -0.8/0.8 and 1 = strong

Question 7

Is the correlation between x and y different from correlation between y and x?

Answer 7

The correlation between x and y is the same as the correlation between y and x

Question 8

What kind of variables does correlation require?

Answer 8

Correlation requires that both variables be quantitative

(cannot compute a correlation between two categorical variables or categorical variable and a quantitative variable)

Question 9

Does correlation change when doing transformations/conversions between units?

Answer 9

Correlation does not change when doing transformations or conversions between units.

This happens because all observations are standardized in the calculation of correlation.

Question 10

What is the unit of correlation?

Answer 10

The correlation (r) has no unit of measurement—it is just a number

Question 11

Interpreting correlation:

What does positive or negative r indicate?

Answer 11

Positive (r) indicates positive association

negative (r) indicates negative association

Question 12

What is the range of correlation?

Answer 12

(r) is always between -1 and 1

• values of r near 0 indicate little/no linear association
• values of r close to -1/1 indicate strong linear association
• r = -1/1 show that points fall exactly on straight line

Question 13

What does correlation only measure?

Answer 13

Correlation only measures the strength of a linear relationship
- curved relationships have a correlation of zero

Question 14

Is correlation affected by outliers?

Answer 14

Correlation is a non-resistant measure (affected by outliers)

Question 15

What are potential reasons for observed association between explanatory and response variable?

Answer 15

1) Causation (best way to establish is through randomized experiment)

2) Confounding variable (there may be causation, but confounding variables make causation hard to prove)

3) Lurking variable (no causation; association can be explained by other variables affecting both explanatory/response)

4) Response variable is causing change in explanatory variable

Question 16

What is a confounding variable?

Answer 16

variable that is not main concern of study but may be partially responsible for the observed results

• causation but tied up with confounding variables

Question 17

What is a lurking variable?

Answer 17

variables that affect both x and y variable, causing us to see an association

• no causation
• other variables affect both explanatory/response

Question 18

What is a regression line?

Answer 18

A straight line that describes how values of a response variable (y) are related on average to values of explanatory variable (x)

• used to estimate average value of y at specific value of x
• used to predict unknown value of y for an individual, given individual’s x value

Question 19

What are components of regression line/equation?

Answer 19

y hat = a + bx

Question 20

What is a?

Question 21

What is b?

Answer 21

slope
- the amount that the y variable changes when x increases by one unit

Question 22

What does it mean if the slope of regression linen is positive? What if it’s negative?

Answer 22

• When slope is positive, direction of relationship is positive (y increases as x increases)
• When slope is negative, direction of relationship is negative (y decreases as x increases

Question 23

What is residual?

Answer 23

observed y value - predicted y value (using regression equation)

-

Question 24

Interpreting residuals:

What does a positive residual mean?

Answer 24

data point falls above regression line

prediction was an underestimate of the observed value

Question 25

Interpreting residuals: What does a negative residual mean?

Answer 25

indicates that data point falls below regression line overestimate of the observed value

Question 26

Why is the regression line we use called least squares regression?

Answer 26

Want a line that comes as close as possible to the points, so we use least squares regression coefficients because they minimize the sum of the squared of the squared residuals

Question 27

What is correlation of determination (r^2)?

Answer 27

the proportion of variation in the y variable explained by the x variable

Question 28

What are characteristics of r^2?

Answer 28

- range between 0 and 1 - high r^2 is better (indicates that a large proportion of the variability in y can be explained by the approximate linear relationship between x and y - near 0 means that x doesn't tell us much about y - near 1 means that x tells us a lot about y

Question 29

How to compute r^2 from r?

Answer 29

square r

Question 30

How to compute r from r^2?

Answer 30

square root r^2

Question 31

What is standard error (Se)?

Answer 31

the typical amount an observation deviates from the least squares regression line - better to have smaller Se as it indicates that residuals tend to be very small (how much accuracy you can expect when using the least squares regression line to make predictions)

Question 32

Locate regression values on Excel output (slope, intercept, r, r 2 , s e)

Answer 32

slope - bottom of coefficients table intercept - intercept or above slope on coefficients table r ~ multiple r r^2 ~ correlation of determination Se - standard error

Question 33

What is the purpose of a residual plot?

Answer 33

a scatter plot of the (x, residual) pairs that can indicate potential problems

Question 34

How does a residual plot evaluate whether a linear regression model is appropriate for the relationship between two variables?

Answer 34

- Curves indicate the data does not follow a linear form - Fanning indicates that residuals are not independent of x-values

Question 35

What is extrapolation?

Answer 35

using a regression line to predict y-values for x-values outside the observed range of the data - making predictions far outside given x values

Question 36

What is the risk of extrapolation?

Answer 36

Riskier the farther we move from the range of the given x-values because there is no guarantee that the relationship will have same trend outside the range of x-values observed