If a particle is dropped at time t 0, physical theory indicates that the relationship between the distance traveled r and the time elapsed t is r gt k for some positive constants g and k. A transformation to linearity can be obtained by taking logarithms:

log r log g k log t.

By letting y log r, A log g, and x log t, this relation becomes y A kx. Due to random error in measurement, however, it can be stated only that E(Yx) A kx. Assume that Y is normally distributed with mean A kx and variance 2

.

A physicist who wishes to estimate k and g performs the following experiment: At time 0 the particle is dropped. At time t the distance r is measured. He performs this experiment five times, obtaining the following data (where all logarithms are to base 10).

1048 20 FORECASTING y log r x log t

3.95 2.0

2.12 1.0 0.08 0.0 2.20 1.0 3.87 2.0

(a) Obtain least-squares estimates for k and log g, and forecast the distance traveled when log t 3.0.

(b) Starting with a forecast for log r when log t 0, use the exponential smoothing method with an initial estimate of log r 3.95 and  0.1, that is, Forecast of log r (when log t 0) 0.1(2.12)

0.9(3.95),

to forecast each log r for all integer log t through log t 3.0.

(c) Repeat part (b), except adjust the exponential smoothing method to incorporate a trend factor into the underlying model as described in Sec. 20.6. Use an initial estimate of trend equal to the slope found in part (a). Let 0.1.