66. Consider a time series—that is, a sequence of observations X1, X2, . . . obtained over time—with observed values x1, x2, . . . , xn. Suppose that the series shows no upward or downward trend over time. An investigator will frequently want to know just how strongly values in the series separated by a specified number of time units are related. The lag-one sample autocorrelation coefficient r1 is just the value of the sample correlation coefficient r for the pairs

(x1, x2), (x2, x3), . . . , (xn1, xn), that is, pairs of values separated by one time unit. Similarly, the lag-two sample autocorrelation coefficient r2 is r for the n  2 pairs (x1, x3),

(x2, x4), . . . , (xn2, xn).

a. Calculate the values of r1, r2, and r3 for the temperature data from Exercise 82 of Chapter 1, and comment.

b. Analogous to the population correlation coefficient , let

1, 2, . . . denote the theoretical or long-run autocorrelation coefficients at the various lags. If all these s are 0, there is no (linear) relationship at any lag. In this case, if n is large, each Ri has approximately a normal distribution with mean 0 and standard deviation 1/n and different Ris are almost independent. Thus H0: i
0 can be rejected at a significance level of approximately

.05 if either ri  2/n or ri  2/n. If n
100 and r1
.16, r2
.09, and r3
.15, is there any evidence of theoretical autocorrelation at the first three lags?

c. If you are simultaneously testing the null hypothesis in part

(b) for more than one lag, why might you want to increase the cutoff constant 2 in the rejection region?

67. A sample of n
500 (x, y) pairs was collected and a test of H0: 
0 versus Ha:  0 was carried out. The resulting P-value was computed to be .00032.