Question

Suppose we have a simple linear regression with following printout.

Coefficients:

Estimate Std. Error t value Pr(>|t|)

(Intercept) -0.06811 0.08375 -0.813 0.421

X 0.86818 0.40886 2.1234 0.046

a. What is the p-value for testing the slope
*H _{0}*

b. Suppose we had a. F-test for adequacy of this regression. What is the value of the test statistic?

What is the p-value?

c. Suppose the sample correlation coefficient of x and y is 0. 852.How much of the variation of y can be

Explained by this regression?

Answer #1

Answer: Suppose we have a simple linear regression with following printout.

Solution:

a) The p-value for testing the slope H0: β1=0 vs. Ha: β1>0:

p-value = 0.046

b) Suppose we had a. F-test for adequacy of this regression, the value of the test statistic:

test statistic = 2.1234

the p-value = 0.046

c. Suppose the sample correlation coefficient of x and y is 0. 852.

r = Corr(x,y) = 0.852

Therefore,

r2 = (0.852)2 = 0.7259

The variation of y can be explained by this regression:

72.59% of the variation of y can be explained by this regression.

The linear regression equation, Y = a +
bX, was estimated. The following computer printout was
obtained:
Given the above information, the value of the
R2 statistic indicates that
0.3066% of the total variation in X is explained by the
regression equation.
30.66% of the total variation in X is explained by the
regression equation.
0.3066% of the total variation in Y is explained by the
regression equation.
30.66% of the total variation in Y is explained by the
regression...

The following table is the output of simple linear regression
analysis. Note that in the lower right hand corner of the output we
give (in parentheses) the number of observations, n, used
to perform the regression analysis and the t statistic for
testing H0: β1 = 0 versus
Ha: β1 ≠ 0.
ANOVA
df
SS
MS
F
Significance F
Regression
1
61,091.6455
61,091.6455
.69
.4259
Residual
10
886,599.2711
88,659.9271
Total
11
947,690.9167
(n = 12;...

In the simple linear regression model estimate Y =
b0 + b1X
A. Y - estimated average predicted value, X –
predictor, Y-intercept (b1), slope
(b0)
B. Y - estimated average predicted value, X –
predictor, Y-intercept (b0), slope
(b1)
C. X - estimated average predicted value, Y –
predictor, Y-intercept (b1), slope
(b0)
D. X - estimated average predicted value, Y –
predictor, Y-intercept (b0), slope
(b1)
The slope (b1)
represents
A. the estimated average change in Y per...

Consider a portion of simple linear regression results,
y^ = 104.93 + 24.73x1; SSE =
407,297; n = 30
In an attempt to improve the results, two explanatory variables
are added. The relevant regression results are the following:
y^ = 4.80 + 19.21x1 –
25.62x2 + 6.64x3; SSE
= 344,717; n = 30.
[You may find it useful to reference the F
table.]
a. Formulate the hypotheses to determine
whether x2 and x3 are
jointly significant in explaining y.
H0:...

8.) Now, do a simple linear regression model for LifeExpect2017
vs. AverageDailyPM2.5. For credit, provide the summary
output for this simple linear regression model.
> Model2 <- lm(LifeExpect2017~ AverageDailyPM2.5)
> summary(Model2)
Call:
lm(formula = LifeExpect2017 ~ AverageDailyPM2.5)
Residuals:
Min 1Q Median 3Q Max
-17.1094 -1.7516 0.0592 1.7208 18.4604
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 81.6278 0.2479 329.23 <2e-16 ***
AverageDailyPM2.5 -0.4615 0.0267 -17.29 <2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘...

R Linear Model Summary. Based on the R output below, answer the
following:
(a) What can infer about β0 and/or β1 ?
(b) What is the interpretation of R2
. (Non-Adjusted) ? In particular, what does it say about how
“x explains y”
(c) Perform the test (α = 0.05): H0 : ρ = 0.5; Ha : ρ > 0.5
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.32632 0.24979 1.306 0.194
x 0.09521 0.01022 9.313 2.93e-15 ***
---
Signif....

In the following regression, X = total assets ($
billions), Y = total revenue ($ billions), and n
= 64 large banks.
R2
0.519
Std. Error
6.977
n
64
ANOVA table
Source
SS
df
MS
F
p-value
Regression
3,260.0981
1
3,260.0981
66.97
1.90E-11
Residual
3,018.3339
62
48.6828
Total
6,278.4320
63
Regression output
confidence interval
variables
coefficients
std. error
t Stat
p-value
Lower 95%
Upper 95%
Intercept
6.5763
1.9254
3.416
.0011
2.7275
10.4252
X1
0.0452
0.0055
8.183
1.90E-11
0.0342
0.0563
(a)...

In a regression analysis involving 27 observations, the
following estimated regression equation was developed. ŷ =
25.2 + 5.5x1 For this estimated
regression equation SST = 1,550 and SSE = 530.
(a) At α = 0.05, test whether
x1 is significant.State the
null and alternative hypotheses.
H0: β1 ≠ 0
Ha: β1 = 0
H0: β0 ≠ 0
Ha: β0 =
0
H0: β0 = 0
Ha: β0 ≠ 0
H0: β1 = 0
Ha: β1 ≠ 0
Find the value...

In a regression analysis involving 27 observations, the
following estimated regression equation was developed.
ŷ = 25.2 + 5.5x1
For this estimated regression equation SST = 1,600 and SSE =
550.
(a) At α = 0.05, test whether
x1is significant.State the null and
alternative hypotheses.
H0: β0 = 0
Ha: β0 ≠ 0
H0: β0 ≠ 0
Ha: β0 =
0
H0: β1 ≠ 0
Ha: β1 = 0
H0: β1 = 0
Ha: β1 ≠ 0
Find the value...

In a regression analysis involving 27 observations, the
following estimated regression equation was developed.
ŷ = 25.2 + 5.5x1
For this estimated regression equation SST = 1,600 and SSE =
550.
(a) At α = 0.05, test whether
x1 is significant.
State the null and alternative hypotheses.
H0: β0 = 0
Ha: β0 ≠
0H0: β0 ≠ 0
Ha: β0 =
0 H0:
β1 ≠ 0
Ha: β1 =
0H0: β1 = 0
Ha: β1 ≠ 0
Find the value of...

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 12 minutes ago

asked 15 minutes ago

asked 17 minutes ago

asked 18 minutes ago

asked 30 minutes ago

asked 34 minutes ago

asked 51 minutes ago

asked 56 minutes ago

asked 56 minutes ago

asked 59 minutes ago

asked 1 hour ago

asked 1 hour ago