# Find a sample size for population in proportion

Can you help me with this question?

Sometimes we perform experiments that compare the probability of success against an external standard, instead of comparing two probabilities. For example, a contract between a supplier and a client can demand that at least $80\%$ of the merchandise will be of grade A. To check whether the supplier stood by the terms of the contract, we can take a small sample of the merchandise and find the sample percentage of grade A, and use that to check the hypothesis that the population percentage is smaller or equal to $80\%$. We will approve the suppliers' work only if $H_0$ will be rejected with $\alpha \ge \text{p-value}$ when $\alpha$ is predetermined.

$p_0$ marks the the percentage of grade A product according to the product. The supplier is convinced that the percentage of merchandise of grade A in the population is $p_0 < p_1$.

Develop a formula for the sample size required so that the percentage required to approve the supplier's work will be $1-\beta$.

I believe I need to make adjustments for this formula: $$n_1 \ge \dfrac{(z_{1- \alpha / 2} \cdot \dfrac{s_0}{s_1} +z_{1-\beta})^2\cdot{s_1}^2}{p_1-p_2}$$

I'm not sure which adjustments though, and how to express the comparison between a proportion and a constant, rather than between two proportions.

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