In a multiple linear regression analysis, (?i , ?1i , ?2i ), ? = 1, … , ? , are statistically independent and satisfy the model (M1) given by:? = ?0 + ?1?1 + ?2?2 + ? ,where the response variable ? is continuous, the regressor vector ? = (?1, ?2)T has mean vector (?x1, ? x2)T and positive definite variance-covariance matrix Σ? , the random errors ?i conditional on ?? are statistically independent and normally distributed with mean zero and variance ?2 which does not depend on ? .a) Consider the case that the value of ?2 is known; that is, it is given.1) Construct a statistical test for testing ?0: ?1 = ?2 = 0, using the known ?2, and the ?-level rejection region of the test. ?0: ?1 = ?2 = 0??: ?1 ≠ ?2 ≠ 02) Construct a statistical test for testing ?0: ?1 = ?2, using the known ?2, and the ?-level rejection region of the test. 3) Are there any differences in testing between a1) and a2) above? why or why not? b) Consider the case that the value of ?2 is unknown; that is, it needs to be estimated.1) Construct a statistical test for testing ?0: ?1 = ?2 = 0 and the ?-level rejection region of the test.2) Construct a statistical test for testing each individual regressor; that is, for the i-th regressor (i = 1, 2), ?0i : ?i = 0 and the ?-level rejection region of the test. 3) In b2) above, derive the probability of falsely rejecting at least one ?0i, ? = 1, 2. In (b1) above, derive the probability of falsely rejecting at least one ?0i, ? = 1, 2. c) If ?0 may or may not be zero in the model M1, construct analysis-of-variance (or sum-of-squares)table for this multiple linear regression analysis. In a multiple linear regression analysis, (y;, x1i , x2;), i = 1, ... ,n , are statisticallyindependent and satisfy the model (M1) given by:y = Bo + B,x1 + B2x2 + E ,where the response variable y is continuous, the regressor vector X = (x1,X2)' has meanvector (ur1, lx2)' and positive definite variance-covariance matrix Ex , the random errorsE conditional on X; are statistically independent and normally distributed with mean zeroand variance ở which does not depend on X.a) Consider the case that the value of o2 is known; that is, it is given.B2 = 0, using the known o², and the1) Construct a statistical test for testing Ho: B1 =a-level rejection region of the test. [5 points]2) Construct a statistical test for testing Ho: B1 = B2, using the known o?, and the a-level rejection region of the test. [5 points]3) Are there any differences in testing between a1) and a2) above? why or why not? [5points]b) Consider the case that the value of o2 is unknown; that is, it needs to be estimated.1) Construct a statistical test for testing Ho: B, = B2 = 0 and the a-level rejectionregion of the test. [5 points]2) Construct a statistical test for testing each individual regressor; that is, for the i-thregressor (i = 1, 2), Hoi: ßi = 0 and the a-level rejection region of the test. [5 points]3) In b2) above, derive the probability of falsely rejecting at least one Hoi , i = 1,2. Inb1) above, derive the probability of falsely rejecting at least one Hoi , i = 1,2. [7 points]c) If Bo may or may not be zero in the model M1, construct analysis-of-variance (or sum-of-squares) table for this multiple linear regression analysis. [3 points]

na multiple linear regression analysis, (y¡, X1i ,X2i), i = 1, ... ,n , are statistically ndependent and satisfy the model (M1) given by: y = Bo + B1x1 + B2X2 + E , here the response variable y is continuous, the regressor vector X = (x1,x,)' has mean ector (ur1, Hr2)' and positive definite variance-covariance matrix Ex, the random errors conditional on X; are statistically independent and normally distributed with mean zero nd variance o which does not depend on X . ) Consider the case that the value of o? is known; that is, it is given. 1) Construct a statistical test for testing Ho: B1 = B, = 0, using the known o², and the a-level rejection region of the test. [5 points] 2) Construct a statistical test for testing Ho: B, = B2, using the known o², and the a- level rejection region of the test. [5 points] 3) Are there any differences in testing between a1) and a2) above? why or why not? [5 points] ) Consider the case that the value of o? is unknown; that is, it needs to be estimated. 1) Construct a statistical test for testing Ho: B, = B2 = 0 and the a-level rejection region of the test. [5 points] 2) Construct a statistical test for testing each individual regressor; that is, for the i-th regressor (i = 1, 2), Hoi: ßi = 0 and the a-level rejection region of the test. [5 points] 3) In b2) above, derive the probability of falsely rejecting at least one Hoi , i = 1,2. In b1) above, derive the probability of falsely rejecting at least one Hoi , i = 1,2. [7 points] If Bo may or may not be zero in the model M1, construct analysis-of-variance (or sum-of- squares) table for this multiple linear regression analysis. [3 points]

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.

Chapter9: Multivariable Calculus

Section9.1: Functions Of Several Variables

Problem 34E: The following table provides values of the function f(x,y). However, because of potential; errors in...

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Question

In a multiple linear regression analysis, (?i , ?1i , ?2i ), ? = 1, … , ? , are statistically independent and satisfy the model (M1) given by:
? = ?0 + ?1?1 + ?2?2 + ? ,
where the response variable ? is continuous, the regressor vector ? = (?1, ?2)T has mean vector (?x1, ? x2)T and positive definite variance-covariance matrix Σ? , the random errors ?i conditional on ?? are statistically independent and normally distributed with mean zero and variance ?2 which does not depend on ? .

a) Consider the case that the value of ?2 is known; that is, it is given.

1) Construct a statistical test for testing ?0: ?1 = ?2 = 0, using the known ?2, and the ?-level rejection region of the test.
?0: ?1 = ?2 = 0
??: ?1 ≠ ?2 ≠ 0

2) Construct a statistical test for testing ?0: ?1 = ?2, using the known ?2, and the ?-level rejection region of the test.

3) Are there any differences in testing between a1) and a2) above? why or why not?

b) Consider the case that the value of ?2 is unknown; that is, it needs to be estimated.

1) Construct a statistical test for testing ?0: ?1 = ?2 = 0 and the ?-level rejection region of the test.

2) Construct a statistical test for testing each individual regressor; that is, for the i-th regressor (i = 1, 2), ?0i : ?i = 0 and the ?-level rejection region of the test.

3) In b2) above, derive the probability of falsely rejecting at least one ?0i, ? = 1, 2. In (b1) above, derive the probability of falsely rejecting at least one ?0i, ? = 1, 2.

c) If ?0 may or may not be zero in the model M1, construct analysis-of-variance (or sum-of-squares)
table for this multiple linear regression analysis.

In a multiple linear regression analysis, (y;, x1i , x2;), i = 1, ... ,n , are statistically
independent and satisfy the model (M1) given by:
y = Bo + B,x1 + B2x2 + E ,
where the response variable y is continuous, the regressor vector X = (x1,X2)' has mean
vector (ur1, lx2)' and positive definite variance-covariance matrix Ex , the random errors
E conditional on X; are statistically independent and normally distributed with mean zero
and variance ở which does not depend on X.
a) Consider the case that the value of o2 is known; that is, it is given.
B2 = 0, using the known o², and the
1) Construct a statistical test for testing Ho: B1 =
a-level rejection region of the test. [5 points]
2) Construct a statistical test for testing Ho: B1 = B2, using the known o?, and the a-
level rejection region of the test. [5 points]
3) Are there any differences in testing between a1) and a2) above? why or why not? [5
points]
b) Consider the case that the value of o2 is unknown; that is, it needs to be estimated.
1) Construct a statistical test for testing Ho: B, = B2 = 0 and the a-level rejection
region of the test. [5 points]
2) Construct a statistical test for testing each individual regressor; that is, for the i-th
regressor (i = 1, 2), Hoi: ßi = 0 and the a-level rejection region of the test. [5 points]
3) In b2) above, derive the probability of falsely rejecting at least one Hoi , i = 1,2. In
b1) above, derive the probability of falsely rejecting at least one Hoi , i = 1,2. [7 points]
c) If Bo may or may not be zero in the model M1, construct analysis-of-variance (or sum-of-
squares) table for this multiple linear regression analysis. [3 points]

Features Features Normal distribution is characterized by two parameters, mean (µ) and standard deviation (σ). When graphed, the mean represents the center of the bell curve and the graph is perfectly symmetric about the center. The mean, median, and mode are all equal for a normal distribution. The standard deviation measures the data's spread from the center. The higher the standard deviation, the more the data is spread out and the flatter the bell curve looks. Variance is another commonly used measure of the spread of the distribution and is equal to the square of the standard deviation.

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