Math 3080-001
Spring 2009
Worksheet #2
1.)
Name: DO NOT TURN THIS IN!
A first step in a regression analysis involving two variables is to construct a
_______________. In such a plot, each (x,y) is represented as a point plotted on a
two-dimensional coordinate system.
2.)
The simple linear regression model is Y = β 0 + β1 x + ε , where the quantity ε is a
random
variable
assumed
to
be
__________
distributed,
with
E (ε ) = __________ and V (ε ) = __________ .
3.)
If S xy = −289.17 and S xx = 340.2, then the least squares estimate of the slope
coefficient β1 of the true regression line y = β 0 + β1 x is βˆ1 = __________.
4.)
If βˆ1 = −1.0, ∑ xi = 10, ∑ yi = 20, and n = 15, then the least squares estimate of the
intercept β 0 of the true regression line y = β 0 + β1 x is βˆ0 = __________.
5.)
The vertical deviations y1 − yˆ1 , y2 − yˆ 2 ,KK , yn − yˆ n from the estimated regression
line are referred to as the __________.
6.)
In a simple linear regression problem, the following statistics are given:
∑x
i
= 10, ∑ xi2 = 55, ∑ xi yi = 130, ∑ yi = 40, ∑ yi2 = 330, βˆ0 = 2.50 and βˆ1 = 1.75,
The error sum of squares is __________.
7.)
In simple linear regression analysis, the __________, denoted by __________,
can be interpreted as a measure of how much variability in y left unexplained by
the model - that is, how much cannot be attributed to a linear relationship.
8.)
In simple linear regression analysis, a quantitative measure of the total amount of
variation in observed y values is given by the _____________________, denoted
by
__________.
9.)
Since the mean of βˆ1 is E ( βˆ1 ) = β1 , then βˆ1 is an __________ estimator of β1 .
10.)
In the simple linear regression model Y = β 0 + β1 x + ε , the quantity ε is a random
variable, assumed to be normally distributed with E( ε ) = 0, and V( ε ) = σ 2 . The
estimated standard error of β̂1 (the least squares estimated of β1 ), denoted by sβˆ ,
1
is ______________ divided by ____________.
11.)
In the simple linear regression model Y = β 0 + β1 x + E , the quantity ε is a random
variable, assumed to be normally distributed with E( ε ) = 0 and V( ε ) = σ 2 . The
estimator β̂1 has a _____________ distribution, because it is a linear function of
independent _____________ random variables.
12.)
The assumptions of the simple linear regression model imply that the standardized
variable T = ( βˆ1 − β1 ) / S βˆ has a t distribution with __________ degrees of freedom.
1
13.)
A 100(1 - α ) % confidence interval for the slope β1 of the true regression line is
β̂1 ± __________ ⋅ sβˆ .
1
14.)
Given that βˆ1 = 1.5, sβˆ = .12 , and n = 15, the 95% confidence interval for the slope
1
β1 of the true regression line (______________,______________).
15.)
In testing H 0 : β1 = 0 versus H a : β1 ≠ 0, the test statistic value is the t – ratio t =
__________ divided by __________.
16.)
In testing H 0 : β1 = 0 versus H a : β1 ≠ 0, using a sample of 15 observations, the
rejection region for .05 level test is either t ≥ __________ or t ≤ __________.
17.)
In testing H 0 : β1 = 0 versus H a : β1 > 0, using a sample of 18 observations, the
rejection region for .025 level test is t ≥ ______________.
18.)
Both the confidence interval for µ y.x , the expected value of Y when x = x∗ , and
∗
prediction interval for a future Y observation to be made when x = x∗ , are
_____________ for an x∗ near x than for an x∗ far from x .
19.)
Let Yˆ = βˆ0 + βˆ1 x∗ , where x∗ is some fixed value of x. Then, the mean value of Yˆ is
E (Yˆ ) = __________.
20.)
A confidence interval refers to a parameter, or population characteristic, whose
value is fixed but unknown to us. In contrast, a future value of Y is not a
parameter but instead a random variable; for this reason we refer to an interval of
plausible values for a future Y as a ____________ rather than a confidence
interval.