We saw in lecture that the sample size needed for a one-sided hypothesis test with specified a and B can be determined by solving the following two equations simulataneously for

n: \frac{k-\mu_{0}}{\sigma / \sqrt{n}}=z_{\alpha} \text { and } \frac{k-\mu_{a}}{\sigma / \sqrt{n}}=-z_{\beta} Show that the required sample size for a one-sided hypothesis test is given by \boldsymbol{n}=\frac{\left(z_{\alpha}+z_{\beta}\right)^{2} \sigma^{2}}{\left(\mu_{a}-\mu_{0}\right)^{2}}