Chapter 4 Elementary Statistics

Front 1

The response variable

Back 1

the variable whose value can be explained by the value of the explanatory or predictor variable.

Front 2

scatter diagram

Back 2

is a graph that shows the relationship between two quantitive variables measured on the same individual. Each individual in the data set is represented by a paint in the scatter diagram. The explanatory variable is plotted on the horizontal axis, and the response variable is plotted on the vertical axis.

Front 3

linear correlation coefficient OR Pearson product moment correlation coefficient

Back 3

is a measure of the strength and direction of the linear relation between two quanitive values. The Greek letter p (rho) represents the population correlation coefficient, and r represents the sample correlation coefficient. We present only the formula of the sample correlation coefficient.

Front 4

The linear correlation coefficient is always between

Back 4

-1 and 1 inclusive. That is -1 ≤ r ≤1

Front 5

if r = +1

Back 5

then a perfect positive linear relation exists between the two variables

Front 6

if r = -1

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the a perfect negative linear relation exists between the two variables

Front 7

if r is close to 0

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then little or no evidence exists of a linear relation between the two variables. SO r CLOSE TO 0 DOES NOT IMPLY NO RELATION, JUST NO LINEAR RELATION.

Front 8

The linear correlation coefficient is a unit less measure of association. So

Back 8

so the unit of measure for x and y plays no role in the interpretation of r.

Front 9

The correlation coefficient is

Back 9

not resistant. Therefore, an observation that does not follow the overall pattern of the data could affect the value of the linear correlation coefficient.

Front 10

positively associated

Back 10

two variables that are linearly related are positively associated. That is two variables are positively associated if, whenever the value of one variable increases, the value of the other variable also increases.

Front 11

negatively associated

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two variables are negatively associated if. whenever the value of one variable increases, the value of the other variable decreases.

Front 12

correlation matrix

Back 12

excel provides it. for every pair of columns in the spreadsheet it will compute and display the correlation in the bottom triangle of the matrix.

Front 13

How do you test for a linear relation

Back 13

1. test for the absolute value of the correlation coefficient.
2. Find the critical value in table 2 from appendix A
3. If the absolute value of the correlation coefficient is greater than the critical value, we say a linear relation exists between the two variables. Otherwise no linear relation exists.

* if the correlation coefficient is positive & greater than the critical value, the variables are positively associated. If the correlation coefficient is negative and less than the opposite of the critical value, the variables are negatively associated.

Front 14

The linear correlation coefficient that implies strong positive or negative association does not imply causation if

Back 14

it was computed using observational data. Why? lurking variables

i.e. as air-conditioning bills increase so does crime rate.

Front 15

xbar

Back 15

sample mean of the explanatory variable

Front 16

sx

Back 16

sample standard deviation of the explanatory variable

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Ybar

Back 17

sample mean of the response variable

Front 18

Sy

Back 18

sample standard deviation of the response variable

Front 19

the closer r is to +1

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the stronger the evidence of positive association

Front 20

The closer r is to -1

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the stronger the evidence of negative association between two variables

Front 21

residual

Back 21

residual represents how close our prediction comes to the actual observation. The smaller the residual the better the prediction.
residual= observed y - predicted y

Front 22

slope (m)

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y₂- y₁ ÷ x₂ - x₁ or rise/run or change in y ÷ change in x

Front 23

slope point formula

Back 23

y - y₁ = m (x - x₁)

Front 24

least-squares regression line

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the line that minimizes the sum of the squared errors (or residuals). This line minimizes the sum of the squared vertical distance between the observed values of y and those predicted by y-hat. We represent this as "minimize ∑residuals²"

Front 25

The equation of the least-squares regression line is given by

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y hat= b₁x + b₀

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a good fit

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means that the line drawn appears to describe the relation between two variables well

Front 27

the slope of the least squares regression line

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b₁ = r × (sy ÷ sx) AND b₀ = ybar - b₁xbar is the y-intercept of the least squares regression line

note: xbar is the sample mean and sx is the sample standard deviation of response variable y

Front 28

in a TI-84 the slope is

Back 28

a

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in a TI-84 the y-intercept is

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b

Front 30

Be careful when using the least-squares regression line to make predictions that are

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much larger or much smaller than those observed

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if the linear correlation between two variables is negative

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the slope of the regression line will also be negative

Front 32

the leas squares aggression line always travels through the point (xbar, ybar)

Back 32

the point (xbar, ybar)

Front 33

residual =

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observed y- predeicted y= y-yhat

Front 34

coefficient of determination (R²)

Back 34

measures the proportion of total variation in the response variable that is explained by the least-squares regression line

in other words, the coefficient of determination is a measure of how well the least-squares regression line describes the relation between the explanatory and response variables. The closer R² is to 1, the better the line describes how changes in the explanatory variable affect the value of the response variable.

Front 35

total deviation

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the deviation between the observed and mean values of the response variable is called the total deviation. So total deviation = y - ybar

total deviation = unexplained deviation + explained deviation

Front 36

explained deviation

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The deviation between predicted and mean values of the response variable is called the explained deviation.

the explained deviation = yhat- ybar

Front 37

unexplained deviation

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deviation between observed and predicted values is the unexplained deviation

unexplained deviation = y-yhat

unexplained variation is also found by summing the squares of the residuals

Front 38

residual plot

Back 38

a scatter diagram with the residuals on the vertical axis and the explanatory variable on the horizontal axis.

Front 39

Constant error variance aka homoscedasticity

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if a plot of the residuals agains the explanatory variables shows the spread of the residuals increasing or decreasing as the explanatory variable increases, then a strict requirement of the linear model is violated.