Plot the vehicle stopping distance versus the speed of travel

Question 1

The distance Y necessary for stopping a vehicle is a function of the speed x of the vehicle. Suppose the following set of data were observed for 12 vehicles travelling at different speeds as shown in the table below.

Vehicle No. Speed, kph Stopping Distance, m
1 40 15
2 9 2
3 100 40
4 50 15
5 15 4
6 65 25
7 25 5
8 60 25
9 95 30
10 65 24
11 30 8
12 125 45

(a) Plot the stopping distance versus the speed of travel.

(b) Assume that the stopping distance is a linear function of the speed, i.e. Y = a + bx + e .
Estimate the regression coefficients, a and b, and the standard deviation sY/x. Also,
determine the correlation coefficient between Y and x.

(c) Determine the 90% confidence interval of the regression equation based on Xi = 9, 30,
60 and 125.

Question 2

The actual concrete strength Y in a structure is generally higher than that measured on a
specimen, x, from the same batch of concrete. Data show that a regression equation for
predicting the actual concrete strength is:
Y = 1.12x + 0.05 (ksi); 0.1 < x < 0.5
and Var(Y ) = 0.0025 (ksi)2

Assume that Y follows a normal distribution for a given value of x.

(a) For a given job, in which the measured strength is 0.35 ksi, what is the probability that
the actual strength will exceed the requirement of 0.3 ksi?

(b) Suppose the engineer has lost the data on the measured strength of the concrete
specimen. However, he recalls that it is either 0.35 or 0.40 with the relative likelihood of
1 to 4. What is the probability that the actual strength will exceed the requirement of 0.3
ksi?

(c) Suppose the measured values of concrete strength at two sites A and B are 0.35 and 0.4
ksi, respectively. What is the probability that the actual strength for the concrete
structure at site A will be higher than that at site B? You may assume that the predicted
actual concrete strength between the sites is statistically independent.

Question 3

Experienced flight instructors have claimed that praise for an unexceptionally fine landing is
typically followed by a poor landing on the next attempt, whereas criticism of a faulty
landing is typically followed by an improved landing. Should we thus conclude that verbal
praise tends to lower performance while verbal criticism tends to raise them? Is some other
explanation possible?