Problem 1 (2.5 points)

Consider the following data on the propagation velocity of an ultrasonic stress

wave through a substance, y (km/s), and the tensile strength of substance, x (MPa).

Table 1: Hypothetical data on the propagation velocity

x, MPa 12 30 36 40 45 57 62 67 71 78 93 94 100 105

y, km/s 3.3 3.2 3.4 3.0 2.8 2.9 2.7 2.6 2.5 2.6 2.2 2.0 2.3 2.1

Suppose a simple linear regression, i.e., y = ßo + ß₁x+ɛ, is used to fit the data,

where E (E) = 0 and Var(e) = o². Least squares estimation is employed to estimate

the model parameters. Compute the following through hand calculation.

(a) What are estimated values for Bo and B₁? What are their corresponding inter-

pretations.

(b) For a two-sided hypothesis test: Ho: B₁ = 0 v.s. H₁: B₁ 0, use p-value

approach and a significance level of α = 0.05 to perform the hypothesis testing

and draw the conclusion.

(c) What is the 95% confidence interval for B₁? What is the corresponding inter-

pretation?

(d) What are values for R2 and ô? If tensile strength of substance is 50 MPa,

what is the predicted propagation velocity of an ultrasonic stress wave through the

substance based on this model?

Note: (i) Use R only when computing the t critical value. Do not use R for the

rest of calculation.

(ii) To get full points, include intermediate steps.