For observed data y=(y1,…,yn)y=(y1,…,yn) with n=21n=21, the above linear regression model was fitted in R, with the following output: >n = 21 >xi = seq(0, n-1,1)/(n-1) >p1 =2*xi-1 >p2 =6*xi^2- 6*xi+1-1/(n-1) > summary(lm(y ~ p1+p2)) Call: lm(formula = y ~ p1 + p2) Residuals: Min 1Q Median 3Q Max -0.5258 -0.2153 0.0813 0.1770 0.4669 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) -0.004238 0.063450 -0.067 0.947 p1 1.181260 0.104784 11.273 1.37e-09 *** p2 -0.953388 0.129422 -7.366 7.77e-07 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 0.2908 on 18 degrees of freedom Multiple R-squared: 0.9097, Adjusted R-squared: 0.8997 F-statistic: 90.68 on 2 and 18 DF, p-value: 3.989e-10 Write this linear regression model in the vector form and answer the following questions, using the above R output where necessary.
For observed data y=(y1,…,yn)y=(y1,…,yn) with n=21n=21, the above linear regression model was fitted in R, with the following output: >n = 21 >xi = seq(0, n-1,1)/(n-1) >p1 =2*xi-1 >p2 =6*xi^2- 6*xi+1-1/(n-1) > summary(lm(y ~ p1+p2)) Call: lm(formula = y ~ p1 + p2) Residuals: Min 1Q Median 3Q Max -0.5258 -0.2153 0.0813 0.1770 0.4669 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) -0.004238 0.063450 -0.067 0.947 p1 1.181260 0.104784 11.273 1.37e-09 *** p2 -0.953388 0.129422 -7.366 7.77e-07 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 0.2908 on 18 degrees of freedom Multiple R-squared: 0.9097, Adjusted R-squared: 0.8997 F-statistic: 90.68 on 2 and 18 DF, p-value: 3.989e-10 Write this linear regression model in the vector form and answer the following questions, using the above R output where necessary.
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter1: Functions
Section1.2: The Least Square Line
Problem 4E
Related questions
Question
For observed data y=(y1,…,yn)y=(y1,…,yn) with n=21n=21, the above linear regression model was fitted in R, with the following output:
>n = 21
>xi = seq(0, n-1,1)/(n-1)
>p1 =2*xi-1
>p2 =6*xi^2- 6*xi+1-1/(n-1)
> summary(lm(y ~ p1+p2))
Call:
lm(formula = y ~ p1 + p2)
Residuals:
Min 1Q Median 3Q Max
-0.5258 -0.2153 0.0813 0.1770 0.4669
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -0.004238 0.063450 -0.067 0.947
p1 1.181260 0.104784 11.273 1.37e-09 ***
p2 -0.953388 0.129422 -7.366 7.77e-07 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.2908 on 18 degrees of freedom
Multiple R-squared: 0.9097, Adjusted R-squared: 0.8997
F-statistic: 90.68 on 2 and 18 DF, p-value: 3.989e-10
Write this linear regression model in the
![3. Consider the following normal linear regression model
where x₂ = (i-1)/(n − 1), i = 1, 2.….., n, and på are Legendre polynomials on [0, 1] given by
Po(x) = 1,
P₁(x) = 2x 1,
P2(x) = 6x² - 6x +1 -1/(n-1).
You are given that
Σ\Po(x;)]²
i=1
a =
E(Yixi)=Bkpk (xi) = ßo + B₁p₁ (xi) + B2p2(xi),
= n₂
22
k=0
Σ\pa(i)]2
i=1
n
ΣPo (xi)P₁ (xi) = 0,
=
n(n+1)
3(n − 1)'
n
EP₂(x)]²=
i=1
n
ΣPo (xi)p2 (xi) = 0,
i=1
(c) Provide estimates for the following values to two decimal places.
(i) Construct a 951% confidence interval [a, b] for 3₁:
=
(n-6)
5
5n²-11n
5(n-1)³
n
ΣP₁ (xi)p2 (xi) = 0.
(12)
i=1
b=
(ii) The estimate of o² is o²
(iii) For testing hypothesis that E(Y₁ | x;) does not depend on x for this model, determine the value of the F
test statistic
and whether this hypothesis is rejected at 51% significance level:
F statistic =
this hypothesis is rejected at 51% significance level: (No answer given) +](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F960e655b-8f45-4427-af2c-42ebd2d45922%2F86f40f08-1fa7-4fd9-83fe-c4e056ce34fb%2Fkz9fpht_processed.png&w=3840&q=75)
Transcribed Image Text:3. Consider the following normal linear regression model
where x₂ = (i-1)/(n − 1), i = 1, 2.….., n, and på are Legendre polynomials on [0, 1] given by
Po(x) = 1,
P₁(x) = 2x 1,
P2(x) = 6x² - 6x +1 -1/(n-1).
You are given that
Σ\Po(x;)]²
i=1
a =
E(Yixi)=Bkpk (xi) = ßo + B₁p₁ (xi) + B2p2(xi),
= n₂
22
k=0
Σ\pa(i)]2
i=1
n
ΣPo (xi)P₁ (xi) = 0,
=
n(n+1)
3(n − 1)'
n
EP₂(x)]²=
i=1
n
ΣPo (xi)p2 (xi) = 0,
i=1
(c) Provide estimates for the following values to two decimal places.
(i) Construct a 951% confidence interval [a, b] for 3₁:
=
(n-6)
5
5n²-11n
5(n-1)³
n
ΣP₁ (xi)p2 (xi) = 0.
(12)
i=1
b=
(ii) The estimate of o² is o²
(iii) For testing hypothesis that E(Y₁ | x;) does not depend on x for this model, determine the value of the F
test statistic
and whether this hypothesis is rejected at 51% significance level:
F statistic =
this hypothesis is rejected at 51% significance level: (No answer given) +
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