Introduction And Abstract Of Research Study 11.2

Front 1

The intercept b0 and slope b1 in the regression model y=b0 +b1x+e
are ______________.

Back 1

population quantities

Front 2

The error variance σ^2ε another ___________ that must be estimated.

Back 2

population parameter

Front 3

The first regression problem is to obtain estimates of the ..

Back 3

slope, intercept, and variance

Front 4

A first step in examining the relation between y and x is to

Back 4

plot the data as a scatterplot.

Front 5

scatterplot: each point in such a plot represents the (x, y) coor- dinates of _____ data entry

Back 5

one

Front 6

The regression analysis problem is to find the best ________ prediction

Back 6

straight-line

Front 7

The most common criterion for “best” is based on squared prediction error. We find the equation of the prediction line—that is, the slope bˆ1 and intercept bˆ0 that minimize the total squared prediction error. The method that accomplishes this goal is called
the_________ because it chooses bˆ0 and bˆ1 to minimize the quantity.

Back 7

least-squares method
∑(yi- yˆi)^2 =∑[yi - (bˆ0 + bˆ1xi)]^2

Front 8

The prediction errors are the __________ from the line

Back 8

vertical deviations. y-yˆ

Front 9

The deviations are taken as vertical distances because we’re trying to predict __ values, and errors should be taken in the __ direction

Back 9

Y, Y

Front 10

____ is the sum of x deviations times y deviations and _____ is the sum of x deviations squared.

Back 10

Sxy, Sxx

Front 11

The estimate of the regression slope can potentially be greatly affected by

Back 11

high leverage points. They carry great weight in the estimate of the slope.

Front 12

These are points that have very high or very low values of the independent variable—outliers in the x direction.

Back 12

high leverage points. They carry great weight in the estimate of the slope.

Front 13

A high leverage point that also happens to correspond to a y outlier is a

Back 13

high influence point. It will alter the slope and twist the line badly.

Front 14

A point has _________ if omitting it from the data will cause the regression line to change substantially.

Back 14

high influence

Front 15

To have high influence, a point must first have _________ and, in addition, must fall _______ the pattern of the remaining points.

Back 15

high leverage , outside

Front 16

If we drew a line through the other points, the line would fall far below this point, so the point is an outlier in the y direction as well. Therefore, it also has __________. Including this point would change the slope of the line greatly

Back 16

high influence

Front 17

the y outlier point corresponds to an x value very near the mean, having low leverage. Including this point would pull the line upward, increasing the intercept, but it wouldn’t increase or decrease the slope much at all. Therefore, it does not have _________.

Back 17

great influence.

Front 18

A high leverage point indicates only a _______ distortion of the equation. Whether or not including the point will “twist’’ the equation depends on its _______ (whether or not the point falls near the line through the remaining points).

Back 18

potential, influence

Front 19

A point must have both _________ and an outlying y value to qualify as a ___________ point.

Back 19

high leverage, high influence

Front 20

Mathematically, the effect of a point’s leverage can be seen in the ____ term that enters into the slope calculation.
∑(xi-x.bar)yi

Back 20

Sxy. the sum of x deviations time y deviations.
We can think of this equation as a weighted sum of y values.

Front 21

Sxy = ∑(xi-x.bar)yi
We can think of this equation as a weighted sum of y values. The weights are ________ positive or negative numbers when the x value is far from its mean and has high leverage. The weight is almost _____ when x is very close to its mean and has low leverage.

Back 21

large, zero.

Front 22

The distinction between __________ (x outlier) and_________ (x outlier and y outlier) points is not universally agreed upon yet

Back 22

high leverage , high influence

Front 23

SE Coef indicates how accurately one can estimate the correct population or process value.
σbˆ1 = σε/√(Sxx)

Back 23

σbˆ1 = σε/√(Sxx)
standard error of the slope bˆ1.
Typically, it is shown in output in a column to the right of the coefficient column. SE Coef

Front 24

The quality of estimation of _______ is influenced by two quantities: the error variance σ^2ε and the amount of variation in the independent variable Sxx.

Back 24

bˆ1
σbˆ1 = σε/√(Sxx)
The standard error of the slope bˆ1

Front 25

The greater the variability σε of the y value for a given value of x, the larger ________ is.

Back 25

The standard error of the slope bˆ1
σbˆ1

Front 26

Sensibly, if there is ___________ around the regression line, it is difficult to estimate that line. Also, the smaller the variation in x values (as measured by Sxx), the ________ σbˆ1 is. σbˆ1 = σε/√(Sxx)

Back 26

high variability, larger
The standard error of the slope bˆ1
σbˆ1 = σε/√(Sxx)

Front 27

The slope is the predicted change in y per unit change in x; if x changes ________ in the data, so that Sxx is small, it is difficult to estimate the rate of change in y accurately.

Back 27

very little.
If the price of a brand of diet soda has not changed for years, it is obviously hard to estimate the change in quantity demanded when price changes.

Front 28

σbˆ1=σε√[1/n+(x.bar^2/(Sxx))]
The standard error of the estimated intercept bˆ0 is influenced by _______, naturally, and also by the _____ of the square of the sample mean, x.bar^2, relative to Sxx.

Back 28

n, size
σbˆ0=σε√[1/n+(x.bar^2/(Sxx))]

Front 29

The _____ is the predicted y value when x = 0; if all the xi are, for instance, _____ positive numbers, predicting y at x = 0 is a huge extrapolation from the actual data. Such extrapolation __________ small errors, and the standard error of bˆ0 is large. The ideal situation for estimating bˆ0 is when x = ______?

Back 29

intercept , large, magnifies, zero

Front 30

σ^2ε:
We can think of this quantity as “variance around the line,’’ or as the

Back 30

mean squared prediction error.

Front 31

The estimate of _____ is based on the ______ (yi - yˆi), which are the prediction errors in the sample.

Back 31

σ^2ε, residuals
mean squared prediction error.

Front 32

The estimate of _______ based on the sample data is the sum of squared residuals divided by n - 2, the degrees of freedom.

Back 32

σ^2ε
mean squared prediction error.

Front 33

s^2ε = [∑(yi-yˆi)^2]/(n-2) = SS(Residual)/ (n-2)
the estimated ______ around the line.

Back 33

s^2ε
variance
The estimated variance is often shown in computer output as MS(Error) or MS(Residual). Recall that MS stands for “mean square’’ and is always a sum of squares divided by the appropriate degrees of freedom.

Front 34

s^2ε = [∑(yi-yˆi)^2]/(n-2) = SS(Residual)/ (n-2)
The reduction from n to n - 2 occurs because in order to estimate the_______ around the regression line, we must first estimate the two parameters____ and ____ to obtain the estimated line. The effective sample size for estimating s^2ε is thus n - 2.

Back 34

variability, b0 , b1

Front 35

In our definition,s^2ε is undefined for n = ______, as it should be.

Back 35

2

Front 36

The square root sε of the sample ______ is called the sample standard deviation around the regression line, the standard error of estimate, or the residual standard deviation.

Back 36

variance

Front 37

Because sε estimates σe, the standard deviation of yi, σe estimates the standard deviation of the _________ of y values associated with a given value of the independent variable x.

Back 37

population

Front 38

ike any other standard deviation, the residual standard deviation may be in- terpreted by the Empirical Rule. About ____ of the prediction errors will fall within 2 standard deviations of the mean error; the mean error is always _____ in the least-squares regression model.

Back 38

95%, 0
Therefore, a residual standard deviation of 0.345 means that about 95% of prediction errors will be less than +/-2(0.345) =0.690.

Front 39

The estimates bˆ0 , bˆ1, and sε are basic in regression analysis. They specify the __________ and the probable degree of ______ associated with y values for a given value of x. The next step is to use these sample estimates to make _____________ about the true parameters.

Back 39

regression line, error, inferences

Front 40

In a MiniTab Output,
What is S?

Back 40

S = sε,the sample standard deviation about the regression line.

Front 41

In a MiniTab Output,
What is MS?

Back 41

MS is s^2ε, SQRT(MS(Residual)) = S = sε
variance
The estimated variance is often shown in computer output as MS(Error) or MS(Residual). Recall that MS stands for “mean square’’ and is always a sum of squares divided by the appropriate degrees of freedom.

Front 42

In a MiniTab Output,
What is SE Coef?

Back 42

SE Coef indicates how accurately one can estimate the correct population or process value.
σbˆ1 = σε/√(Sxx)

Front 43

The _______ around the fitted line (the residual standard deviation) is shown as S = 2.72162. Therefore, about 95% of the prediction errors should be less than

Back 43

standard deviation
S = sε,the sample standard deviation about the regression line.