Consider a time series data set of size 50 with $x_t = \frac{1}{3}(z_t + z_{t-1} + z_{t-2})$ process, where $z_t \sim WN(0, \sigma^2)$.\\
$\gamma_k = \begin{cases} \frac{1}{3}\sigma^2, & k = 0 \\ \frac{2}{9}\sigma^2, & k = 1 \\ \frac{1}{9}\sigma^2, & k = 2 \\ 0, & \text{Otherwise} \end{cases}$, $\rho_k = \begin{cases} \frac{3-k}{3}, & k = 0, 1, 2 \\ 0, & \text{Otherwise} \end{cases}$, $\bar{x}_t = 0.157$ and $\sigma^2 = 1$.\\a) Calculate $var(\bar{x})$.\\b) Construct an approximate 95\% confidence interval for $\mu$.\\c) Is the data set computable with the hypothesis that $\mu = 0$? Justify your answer.\\Hint: One of the following values might be useful: $z_{0.025} = 1.645$ or $z_{0.025} = 1.960$.
