Consider a normal population with unknown mean $\mu$ and variance $\sigma^2=9$. To test $H_{0}:\mu=0$,against $H_{1}:\mu \ne 0$. A random sample of size 100 is taken. Based on this sample, the test of the form $|\bar{X_n}| > K$ rejects the null hypothesis at 5% level of significance. Then, which of the following is a possible 95% confidence interval for $\mu$ ?
(A) (-0.488, 0.688) (B) (-1.96, 1.96) (C) (0.422, 1.598) (D) (0.588, 1.96)
Now, if we construct the test statistic $\tau=\frac{\bar{X_n}-\mu}{0.3} \sim N(0,1)$ Therefore our test ,actually based on the observed value of $\tau=\tau_{0}$ is given by $|\tau_{0}|> \tau_{\frac{\alpha}{2}}$ where $\alpha=0.05$ Now,we get $K=\frac{3}{10}\tau_{0.025}=0.588$ Thus the 95% confidence interval for $\mu$ is coming to be $(\bar{X_n}-0.588 ,\bar{X_n}+0.588)$So, looking at the options it seems that both (A) , (C) are correct but only (C) is given to be correct. Please help!