In an experiment to estimate the acceleration of an object down an inclined plane, the object is released and its distance in meters (y) from the top of the plane is measured every 0.1 second from time t = 0.1 to t = 1.0. The data are presented in the following table.
t y
0.1....................0.03
0.2.....................0.1
0.3....................0.27
0.4....................0.47
0.5....................0.73
0.6....................1.07
0.7....................1.46
0.8....................1.89
0.9....................2.39
1.0....................2.95
The data follow the quadratic model y = β0 + β1t + β2t2 + ε, where β0 represents the initial position of the object, β1 represents the initial velocity of the object, and β2 = a/2, where a is the acceleration of the object, assumed to be constant. In a perfect experiment, both the position and velocity of the object would be zero at time 0. However, due to experimental error, it is possible that the position and velocity at t = 0 are nonzero.
a. Fit the quadratic model y = β0 + β1t + β2t2 + ε.
b. Find a 95% confidence interval for β2.
c. Find a 95% confidence interval for the acceleration a.
d. Compute the P-value for each coefficient.
e. Can you conclude that the initial position was not zero? Explain.
f. Can you conclude that the initial velocity was not zero? Explain.