Suppose that Y1, Y2, . . . , Yn are independent, normally

Chapter 11, Problem 28E

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QUESTION:

Problem 28E

Suppose that Y1, Y2, . . . , Yn are independent, normally distributed random variables with E(Yi) = β0 + β1 xi and V (Yi ) = σ 2, for i = 1, 2, . . . , n. Show that the likelihood ratio test of H0 : β1 = 0 versus Ha : β1 ≠ 0 is equivalent to the t test given in this section.

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QUESTION:

Problem 28E

Suppose that Y1, Y2, . . . , Yn are independent, normally distributed random variables with E(Yi) = β0 + β1 xi and V (Yi ) = σ 2, for i = 1, 2, . . . , n. Show that the likelihood ratio test of H0 : β1 = 0 versus Ha : β1 ≠ 0 is equivalent to the t test given in this section.

ANSWER:

Step 1 of 10

The density function of the random variable

where

Because the random variables  are independent, the likelihood function is equal to:

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