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 ϵ 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 2000? (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 2500? (Round your answer to four decimal places.)

(c)

Consider making two independent observations on 28-day strength, the first for an accelerated strength of 2000 and the second for

x = 2500.

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.)

Answer 1

Given that y = 1900 + 1.4x and sd = 350

a.

probability that the observed value of 28-day strength will exceed 5000 psi when the value of accelerated strength is 2000.

y = 1900 + 1.4 * 2000 = 4700

P(x > 5000) = P( z > ((5000 - 4700)/350)

=> P(z > 0.857)

=0.1957

b.

y = 1900 + 1.4 * 2500 = 5400

P(X > 5000) = P(Z > ((5000 - 5400)/350)

=P( Z > -1.142857)

=0.8735

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

E(Y2 -Y1)= 1.3 *(2500 - 2000) = 650

P((y2 -Y1) > 1000) = P( Z > (1000 - 650)/sqrt(var(y2 -y1))

P(Z > (1000 - 650)/245000)

P(Z > 0.707)

0.2389

d.

sqrt(Var(Y2 -Y1)) = sqrt(245000) = 494.97

E(Y2 - Y1) = (1800 + 1.3 X2 - (1800+1.3X1))

=1.3(X2 - X1)

P(Y2 > Y1) = 0.95

P(Y2 -Y1>0) = 0.95

we have P(Z > -1.64) = 0.95

therefore -1.3(x2 - x1)/494.97 = -1.645

x2 -x1 = 626.3274