An article considered regressing y = 28-day standard-cured strength (psi) agains

Question

An article considered regressing

y = 28-day standard-cured strength (psi) against x = accelerated strength (psi). Suppose the equation of the true regression line is y = 1900 + 1.4x, and that the standard deviation of the random deviation is 350 psi.

(a)

What is the probability that the observed value of 28-day strength will exceed 5000 psi when the value of accelerated strength is 2100? (Round your answer to four decimal places.)

(b)

What is the probability that the observed value of 28-day strength will exceed 5000 psi when the value of accelerated strength is 2600? (Round your answer to four decimal places.)

(c)

Consider making two independent observations on 28-day strength, the first for an accelerated strength of 2100 and the second for x = 2600.

What is the probability that the second observation will exceed the first by more than 1000 psi? (Round your answer to four decimal places.)

(d)

Let Y1 and Y2 denote observations on 28-day strength when x = x1 and x = x2, respectively. By how much would x2 have to exceed x1 in order that P(Y2 > Y1) = 0.95? (Round your answer to two decimal places.)

Explanation / Answer

We have given y^ = 1900 + 1.4x ,and standard deviation = 350 .

a) We have to find probability that the observed value of 28-day strength will exceed 5000 psi when the value of accelerated strength is 2100.

y^ = 1900 + 1.4*2100 = 4840

We have to find P( x > 5000 ) =   (5000 - 4840 ) / 350 = 0.46

Using Z score table we have to find probability for Z = 0.46

P( Z > 0.46 ) = 1 - P( Z < 0.46 ) = 1 - 0.6672 = 0.3228

b) We have to find probability that the observed value of 28-day strength will exceed 5000 psi when the value of accelerated strength is 2600 .

y^ = 1900 + 1.4*2600 = 5540

We have to find P( x > 5000 ) =   (5000 - 5540 ) / 350 = -1.54

Using Z score table we have to find probability for Z = -1.54

P( Z > -1.54 ) = 1 - P( Z < -1.54 ) = 1 - 0.0618 = 0.9382.

c) We have to consider making two independent observations on 28-day strength, the first for an accelerated strength of 2100 and the second for x = 2600.

And we have to find probability that the second observation will exceed the first by more than 1000 psi

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