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Chpt 12 - 20 > chpt 14 > Flashcards

chpt 14 Flashcards

1
Q

What does simple linear regression use

A
  1. one independent variable and one dependent variable

2. uses a straight line to approximate the relationship

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2
Q

What does multiple regression use

A
  • 2 or more independent variables
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3
Q

What are the 2 objectives for simple linear regression

A
  1. establish if there is a relationship b/w 2 variables (ie income and spending)
  2. Forecast new observations (ie. sales over next Qrt)
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4
Q

What is the dependent variable denoted by

A

y

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5
Q

what is the independent variable denoted by

A

x

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6
Q

what is the dependent variable

A

the variable being predicted

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7
Q

what is the independent variable

A

the variable(s) used to predict the values of the dependent variable

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8
Q

What is the formula for the simple linear regression

A

Y = B0 + B1X + E

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9
Q

What does Y represent in the simple linear regression model

A

the dependent variable

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10
Q

what does B0 represent in the simple linear regression model

A

intercept or constant

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11
Q

what does B1x represent

A

coefficient of x or slope of x

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12
Q

what does E represent in the simple linear regression model

A

error term

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13
Q

What does the error term account for in the simple linear regression model

A

accounts for the variability in y that can’t be explained by the linear relationship b/w x and y

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14
Q

What is the simple linear regression equation

A

E(y) = B- + B1x

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15
Q

What does E(y) represent

A

mean or expected value of y for a given value of x

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16
Q

What can we note about B0 adn B1 in the simple linear regression equation

A

they are known

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17
Q

What is the estimate simple linear regression equation

A

y hat = b0 + b1x

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18
Q

When do we use the estimate simple linear regression equation

A

when B0 and B1 are NOT known

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19
Q

What does y hat represent in the estimate simple linear regression equation

A

point estimate of E(y)

- provides a prediction of an individual value of y for a given value of x

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20
Q

What are B0 and B1

A

population parameters

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21
Q

what are b0 and b1

A

sample statistics to estimate B0 and B1

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22
Q

IF we are trying to predict sales for a given level of advertising what is the dependent and independent variable

A

Dependent variable - sales (y)

Independent variable - advertising expenditures (x)

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23
Q

what does “simple” indicate in simple linear regression

A

one independent variable and one dependent variable

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24
Q

What does “linear” Indicate in Simple linear regression

A

the relationship is approximated using a straight line

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25
Q

What is B0 in the simple linear regression model

A

the y-intercept of the regression line or the value of y when x is 0

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26
Q

What is B1 in the simple linear regression model

A

the slope of the regression line

  • the line tells us two things
    1. whether the line is increasing or decreasing
    2. how steep it is
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27
Q

What is E in the simple linear regression model

A

the error term

- as good as our model might be, there is always random error term that cannot be accounted for

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28
Q

if the line slopes upward, what is the relationship

A

as x increases, so does y - positive relationship

B1 - will be positive

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29
Q

if the line slopes downward, what is the relationship

A

as x increases, y decreases, negative relationship

B1 - would be negative

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30
Q

what if the line is straight across (the regression line is flat)

A

no relationship, as x increases, y remains the same

B1 is 0

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31
Q

What are the POPULATION parameters for the y intercept and the slope

A

B0 and B1

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32
Q

what are the sample statistics used to estimate B0 and B1

A

b0 and b1

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33
Q

what does y hat represent in the simple linear regression

A

the predicted value of y for a given x value

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34
Q

what is the estimated simple linear regression equation

A

y hat = b0 +b1x

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35
Q

What does the Coefficient of Determination tell us about the estimated regression equation

A

how well does the estimated regression equation fit the data

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36
Q

What does the Coefficient of Determination provides us with

A

a measure of the goodness of fit

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37
Q

in Coefficient of determination, what is the ith residual

A

the predicted value of the dependent variable y hat i

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38
Q

for the ith observation, the residual is indicated by what

A

yi- y hat i

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39
Q

What is the formula for the coefficient of determination

A

r squared = SSR/SST

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40
Q

what does r squared represent

A

the coefficient of determination

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41
Q

What does SSR stand for in coefficient of determination

A

sum of squares due to regression

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42
Q

what does SST stand for in Coefficient of determination

A

sum of squares for the total deviation

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43
Q

What is the formula for SSR in coefficient of determination

A

sum (y hat i - y bar) squared

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44
Q

what does the SSR in coefficient of determination measure

A

the difference b/w the predicted values and the average or

how much the y hat values on teh estimated regression line deviates from y hat

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45
Q

What does SSE in coefficient of determination stand for

A

sum of squares due to Error

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46
Q

What is the formula in Coefficient of determination for SST

A

sum (yi - ybar) squared

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47
Q

what is the formula in Coefficient of determination for SSE

A

sum (yi - y hat i) squared

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48
Q

In Coefficient of determination, how do you calculate SST

A

SST = SSR + SSE

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49
Q

What should we expect regarding SST, SSR and SSE in the coefficient of determination

A

we should expect that SST, SSR and SSE related from

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50
Q

What would be a perfect fit in coefficient of determination

A

SSR = SST

SSR / SST = 1

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51
Q

What would a poor fit be in coefficient of determination

A

large values for SSE

- poorest fit when SSR = 0 and SSE = SST

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52
Q

What is r squared

A

percent of variability in y can be explained by x

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53
Q

if r squared = 95.5%, what can we say

A

95.5% of the variability in grades for instance, can be explained by the number of hours studied

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54
Q

What does the correlation Coefficient measure

A

it measures the strength of association b/w x and y

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55
Q

What does the correlation Coefficient measure

A

it measures the strength of association b/w x and y

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56
Q

what is the correlation Coefficient denoted by

A

r

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57
Q

what are the values of r in correlation Coefficient

A

between -1 and +1

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58
Q

In Correlation Coefficient, if r = 1, what does this mean

A

means perfect positive linear relationship b/w x and y

  • no deviation
  • all the data points from the sample lay exactly on the line of regression with no deviation and the line slopes upward
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59
Q

In Correlation Coefficient, if r = -1, what does this mean

A

means perfect negative linear relationship b/w x and y

  • no deviation
  • all data points from the sample lay exactly on the line of regression with no deviation and the line slopes downward
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60
Q

In Correlation Coefficient, if r = 0, what does this mean

A

no relationship b/w x and y

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61
Q

what is the formula for correlation Coefficient

A

rxy = (sign of b1)x square root of coefficient of determination

or
rxy = (sing of b1) x square root of rsqaured

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62
Q

in correlation Coefficient, what is b1

A

slope of the estimate

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63
Q

In correlation coefficient, since the square root of anything doesn’t tell us if the number was negative or postive we have to look at what

A

the slope and then we use the sign for our slope

example b1 is positive 4.74 then we use positive sign
rxy = +.9505

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64
Q

if rxy is .9749 what does this indicate

A

a very strong positive linear relationship bw x and y

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65
Q

Testing for Significance if y=B0+B1x +E

if B1 = 0 then Y=

A

B0 no matter what value x is
- the value of y does not depend on x
(no linear relationship b/w x and y)

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66
Q

What is the null hypothesis and the alternative for testing significance in Simple Linear Regression

A 
Ho= B1  = 0
Ha = B1 does not = 0
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67
Q

What test do we use when testing for signfiicanace in simple linear regression

A

t test

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68
Q

what is the formula for the test statitistic when testing for significance in simple linear regression

A

t = b1 / sb1

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69
Q

what does sb1 stand for

A

standard error for slope

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70
Q

what is the formula for sb1 (the standard error for the slope)

A

sb1 = s (standard deviation) / square root sum (xi - xbar)squared

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71
Q

what is the formula for s in sb1

A

s = square root of (SSE/n-2)

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72
Q

Coefficient of determination - Definition

A

A measure of the goodness of fit of the estimated regression equation. It can be interpreted as the proportion of the variability in the dependent variable y that is explained by the estimated regression equation

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73
Q

Confidence interval - Definition

A

The interval estimate of the mean value of y for a given value of x.

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74
Q

Correlation coefficient - Definition

A

A measure of the strength of the linear relationship between two variables

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75
Q

Dependent variable - definition

A

The variable that is being predicted or explained. It is denoted by y.

76
Q

Estimated regression equation - Definition

A

The estimate of the regression equation developed from sample data by using the least squares method. For simple linear regression, the estimated regression equation is yˆ = b0 + b1x.

77
Q

High leverage points - Definition

A

Observations with extreme values for the independent variables

78
Q

Independent variable - Definition

A

The variable that is doing the predicting or explaining. It is denoted by x.

79
Q

Influential observation - Definition

A

An observation that has a strong influence or effect on the regression results.

80
Q

ith residual - Definition

A

The difference between the observed value of the dependent variable and the value predicted using the estimated regression equation; for the ith observation the ith residual is yi − yˆi.

81
Q

Least squares method - Definition

A

A procedure used to develop the estimated regression equation. The objective is to minimize o( yi − yˆi)2.

82
Q

Mean square error - Definition

A

The unbiased estimate of the variance of the error term s2. It is denoted by MSE or s2.

83
Q

Normal probability plot - Definition

A

A graph of the standardized residuals plotted against values of the normal scores. This plot helps determine whether the assumption that the error term has a normal probability distribution appears to be valid

84
Q

Outlier - Definition

A

A data point or observation that does not fit the trend shown by the remaining data

85
Q

Prediction interval - Definition

A

The interval estimate of an individual value of y for a given value of x.

86
Q

Regression equation - Definition

A

THe Equation that describes how the mean or expected value of the dependent variable is related to the independent variable; in simple linear regression, E(y) = b0 + b1x.

87
Q

Regression model - Definition

A

The equation that describes how y is related to x and an error term; in simple linear regression, the regression model is y = b0 + b1x + e.

88
Q

Residual Analysis - Definition

A

The analysis of the residuals used to determine whether the assumptions made about the regression model appear to be valid. Residual analysis is also used to identify outliers and influential observations

89
Q

Residual Plot - Definition

A

Graphical representation of the residuals that can be used to determine whether the assumptions made about the regression model appear to be valid

90
Q

Scatter Diagram - Definition

A

A graph of bivariate data in which the independent variable is on the horizontal axis and the dependent variable is on the vertical axis

91
Q

Simple linear regression - Definition

A

Regression analysis involving one independent variable and one dependent variable in which the relationship between the variables is approximated by a straight line.

92
Q

Standard error of the estimate - Definition

A

The square root of the mean square error, denoted by s. It is the estimate of s, the standard deviation of the error term e

93
Q

Standardized residual - Definition

A

The value obtained by dividing a residual by its standard deviation

94
Q

Regression and Correlation analysis are used to study what

A

relationships between two or more variables

95
Q

The focus in correlation analysis is on assessment of

A

the size and the direction of the relationship

96
Q

The relationship between variables is said to be positive if

A

the two variables increase together and decrease together

97
Q

The relationship between variables is said to be negative if

A

they move in different directions

98
Q

In regression analysis, on the other hand, the focus is on what

A

prediction

99
Q

The value of one variable is predicted form the value of another variable based on what

A

a model relating the two variables. the model has to be estimated, using a sample from the bivariate distribution, before it can be used

100
Q

What is the first thing to do in a regression analysis

A

plot the data as a scatter diagram, and see if the assumption of a linear relationship is plausible

101
Q

If the data points are scatter in such a way that a straight line can be drawn through them they are

A

clustered around the line and the assumption of a linear relationship is reasonable

102
Q

if the data is scattered in such a way that a straight line cannot be drawn through them then they are

A

such as a curve, or in no pattern at all the assumption is violated that it is a linear relationship

103
Q

Geometrically, the actual value yi is the ________ of the point form the _________axis to the point on the regression line

A

height

horizontal

104
Q

The distance between the two (yi and Y triangle hat) is the

A

error at the given xi

105
Q

if the actual value of y triangle hat is on the regression line, then yi ________ and the error is ________

A

= y triangle hat and the error is zero

106
Q

If the actual value yi-y triangle hat is above the regression line, ,this results in a _________

A

positive error

107
Q

if the actual value of yi-y triangle hat is below the regression line, this results in a

A

negative error

108
Q

Is there an error term for each data point?

A

yes

109
Q

When does a perfect fit occur in the simple linear regression

A

when all these error terms are zero, which means all the data points are lined up along a straight line

110
Q

is a perfect fit rare in simple linear regression

A

yes

111
Q

what would be the best regression line

A

is a line throughthe points that minimizes the errors in some sense

112
Q

Who proposed the least squares method

A

Gauss

113
Q

Explain the least squares method

A

deals with the squares of the errors, instead of the errors themselves, so it treats only the size of the errors and not heir sings

114
Q

What does the least squares method do

A

it minimizes the sum of squared errors

115
Q

Why is the relation of SST = SSR + SSE fundamental

A

because it assesses the contribution of the regression as a source of variation compared to other sources of variation in the data

116
Q

since we are studying the effect of regression alone, all other sources of variation are what

A

lumped under the general label “error” and are treated as one source. This is similar to the notion of “between” and “within” variations

117
Q

What is the mean Square due to error for simple linear regression

A

one measure of the goodness of fit for a regression equation

118
Q

MSE is also

A

S^2

119
Q

MSE is useful

A

only in a relative sense

a value of say, 13.829 for MSE does not tell us whether the fit is good or bad. nor, if good, does it tell us how good the fit is.

It is only useful when we compared it with MSE for another model or fit

120
Q

When comparing MSE which one is better

A

the one with the smaller MSE is better

121
Q

What is MSE very sueful for

A

constructing tests of significance and confidence intervals

122
Q

What is the square root of MSE use for

A

to estimate the standard error of an estimate

which servers as a benchmark for decisions regarding the size of a difference between an estimate and its hypothesized value

123
Q

What test is used with MSE for Simple linear regression

A

t-test

124
Q

Can MSE be used as a comparison by itself and what test is used

A

a comparison can be made directly with the MSE, since MSE is itself a measure of variation. This results in the F test ?

125
Q

An observation can be both an outlier and what

A

an influential observation, it can be an outlier but not an influential observation, or it can be an influential observation but not an outlier

126
Q

In identifying an outlier, we focus on what

A

the y value (or equivalently, on the residual or standardized residual) of a point

127
Q

When identifying an influential observation, the focus is on what

A

the x values

128
Q

Observations who x values are very different from the x values of the rest of the data are most likely

A

influential observations

129
Q

Those whose y values are way off the trend of the other points are most likely

A

outliers

130
Q

The variable being predicted is called

A

the dependent variable

131
Q

What are the independent variable

A

The variable or variables being used to predict the value of the dependent variable are called the independent variables

132
Q

In simple linear regression, each observation consists of two values, what are they

A
  1. one for the independent variable

2. one for the dependent variable

133
Q

Can regression analysis be interpreted as a procedure for establishing a cause-and-effect relationship between variables?

A

no, it can only indicate how or to what extent variables are associated with each other

any conclusions about cause and effect must be based upon the judgement of those individuals most knowledgeable about eh application

134
Q

Using the estimated regression equation to make predictions outside the range of the values of the independent variable - what caution is there

A

should be done with caution because outside that range we cannot be sure that the same relationship is valid

135
Q

What does the least squares method provides for the estimated regression equation do

A

minimizes the sum of squared deviations between the observed values of the dependent variable yi and the predicated values of the dependent variable ytraingle hat i

136
Q

The least squared criterion for estimated regression is used to do what

A

to choose the equation that provides the best fit. It is the mostly widely used method

137
Q

Coefficent of determination provides what

A

a measure of goodness of fit for the estimated regression equation

138
Q

What is SSE a measure of in the estimated regression

A

it is a measure of the error in using the estimated regression equation to predict values of the dependent variable in the sample

139
Q

If you don’t have the knowledge of the xi, what would you use to estimate of something

A

you would use the mean value

the estimated regression is a much better predictor than using the mean value

140
Q

What can SSR be thought of as

A

the explained portion of SST

141
Q

What can SSE be thought of as

A

the unexplained portion of SST

142
Q

What would be a perfect fit for the estimated regression

A

yi - y triangle hat = 0

this means that every value of the dependent variable yi lies on the estimated regression line

143
Q

If the estimate regression is a perfect fit, what can we say about SSE

A

SSE = 0

and SSR/SST = 1

144
Q

If SSE is large, what can we say about the estimated regression

A

poorer fits will have larger values of SSE

145
Q

What would be the poorest fit when

A

the largest value for SSE occurs when SSR = 0 and SSE=SST

146
Q

When SSE = SST what kind of fit is this

A

poorest fit

147
Q

What values will the ratio SSR/SST take

A

take on teh values between 0 and 1

148
Q

what does r^2 stand for

A

coefficient of determination

149
Q

r^2 formula

A

SSR/SST

150
Q

if r^2 SSR/SST =close to 1

A

good a fit

151
Q

What would r^2 = .9027 mean

A

90.27% of the variability in yi can be explained by the estimated regression equation

152
Q

What is correlation Coefficient a measure of

A

a descriptive measure of the STRENGTH of linear association between two variables (x and y)

153
Q

What are the values that correlation coefficient take on

A

between -1 and +1

154
Q

What does a value of +1 Correlation Coefficient mean

A

indicates that the two variables x and y are perfectly related in a positive linear sense. That all data points are on a straight line that has a positive slope

155
Q

What do values close to zero represent for Correlation Coefficent

A

indicate that x and y are not linearly related

156
Q

What is the formula for Correlation Coefficient

A

rxy = (sing of b1) Square root of Coefficient of determination

rxy = square root of r^2

157
Q

Coefficient of determination provides a measure between what numbers

and

Correlation Coefficient provides a measure between what numbers

A

Coefficient of Determination
r^2 is between 0 and 1

Correlation Coefficient
rxy = square root of r^2 is between -1 and +1

158
Q

The sample correlation coefficient is restricted to what

A

A linear relationship between two variables

159
Q

The coefficient of determination can be used for what

A

nonlinear relationships and for relationships that have two or more intendent variables

thus, the coefficient of determination r^2, provides a wider range of applicability

160
Q

Which provides a wider range of applicability coefficient of determination or Correlation Coefficient

A

Coefficient of Determination

161
Q

When using r^2 we can draw no conclusion about what

A

whether the relationship between x and y is statistically significant - such conclusion must be based on considerations that involve the sample size and the properties of the appropriate sampling distributions of the least squares estimators

162
Q

SSE is what in Simple linear regression

A

sum of squared residuals

163
Q

SSE is a measure of what

A

of the variability of the actual observations about the estimated regression line

164
Q

In simple linear regression, does the F test and t test provide the same results?

A

yes, if it is just for one I.V.

165
Q

If it is more than one IV, does the F test and t test provide the same results

A

no, only the F test can be used to test for an overall significant relationship

166
Q

Confidence intervals and prediction intervals show the precision of the regression results. Narrower intervals provide what

A

a higher degree of precision

167
Q

A confidence interval is an interval estimate of what

A

the mean value of y for a given value of x

168
Q

a prediction interval is an interval estimate of what

A

used to predict an individual value of y for a new observation corresponding to a given value of x

169
Q

the margin of error is large for which interval, a confidence interval or prediction interval

A

prediction interval

170
Q

What is the margin of error associated with a prediction interval

A

t a/2 spread

171
Q

In general, the lines of the confidence interval limits and the prediction interval limits both have what

A

curvature

172
Q

confidence intervals and prediction intervals are both more precise when the value of the IV x* is closer to

A

x bar

173
Q

What may an outlier represent

A
  1. erroneous data - error recording, s/b corrected
  2. signal a violation of the model assumption - may need to consider another model
  3. unusual values that occurred by chance - should stay
174
Q

What is an influential observation

A
  1. it could be an outlier
  2. can influence how the data is interpreted

if this data set was removed, it would change our slope from negative to positive for example

175
Q

if the Influential observation is valid

A
  1. can contribute to a better understanding of the appropriate mode and lead to a better estimate regression equation
  2. try to obtain data on intermediate values of x to better understand the relationship b/w x and y
176
Q

What is high leverage

A

the father xi is form it’s mean (x bar) the higher the leverage of the observation

  • need computer software to help with this
177
Q

Explain lower leverage

A

outside of the other data sets but won’t change the line

178
Q

explain high leverage

A

outside of the other data sets by a lot

179
Q

explain lower leverage low influence

A

near to the line

180
Q

explain high leverage low influence

A

need some work

181
Q

HOw is outlier determined

A

if it is outside of the +2 or -2 from the mean line

182
Q

What is an outlier denoted as on a computer print out

A

R

183
Q

if we only have one variable, how can we predict another amount

A

by using the mean

184
Q

if we are only using one variable to predict, what is the best fit line

A

the mean

185
Q

What do you use to measure of how well the estimated regression line FITS the data

A

R2 the coefficient of determination

186
Q

What test do we use to test whether B1 is significant

A

t-test b1/sb1

Chpt 12 - 20 (10 decks)

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