Using the $C$% confidence interval for $\mu$ to perform one sample $t$ test

Be sure you understand the theory behind confidence intervals and significance tests, before trying to understand the explanation behind the 'explain' button.
Two sided test If $\mu_0$ is not in the $C$% confidence interval, the sample mean is significantly different from $\mu_0$ at significance level $\alpha = 1 - C/100$.
For instance, if $\mu_0$ is not in the $95$% confidence interval, the sample mean is significantly different from $\mu_0$ at significance level $\alpha = 1 - 95/100 = .05$.
Explain
Right sided test If $\mu_0$ is not in the $C$% confidence interval and the sample mean is larger than $\mu_0$, the sample mean is significantly larger than $\mu_0$ at significance level $\alpha = \frac{1 \, - \, C/100}{2}$.
For instance, if $\mu_0$ is not in the $90$% confidence interval and the sample mean is larger than $\mu_0$, the sample mean is significantly larger than $\mu_0$ at significance level $\alpha = \frac{1 \,-\, 90/100}{2} = .05$.
Explain
Left sided test If $\mu_0$ is not in the $C$% confidence interval and the sample mean is smaller than $\mu_0$, the sample mean is significantly smaller than $\mu_0$ at significance level $\alpha = \frac{1 \,-\, C/100}{2}$.
For instance, if $\mu_0$ is not in the $90$% confidence interval and the sample mean is smaller than $\mu_0$, the sample mean is significantly smaller than $\mu_0$ at significance level $\alpha = \frac{1 \,-\, 90/100}{2} = .05$.
Explain