Suppose {$Z_t$} is white noise with mean zero. Define the stochastic process {$X_t$}, $t=1,2,3\ldots$ by $$X_t = \alpha X_{t-1} + Z_t.$$ Consider a sample $X_1 ,X_2 ,$ $\dots ,$ $X_{200}$ taken from {$X_t$} on 200 consecutive time points. Let the mean of these observations be $\bar{X}$.

Part (a) Suppose $\alpha =$. Compared to the sample mean of a sample of size 200 taken at random from a random variable with the same variance as {$X_t$}, $\bar{X}$ will have

A. the same variance.
B. lower variance.
C. higher variance.

Part (b) Compared to the sample mean of a sample of size 100 taken at random from a random variable with the same variance as {$X_t$}, $\bar{X}$ will have

A.
B.
C.

Part (c) Now suppose $\alpha =$. Compared to the sample mean of a sample of size 200 taken at random from a random variable with the same variance as {$X_t$}, $\bar{X}$ will have

A. higher variance.
B. lower variance.
C. the same variance.