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**Robb** A manufacturer claimed that at least 20% of the public preferred her product. A sample of 100 persons is taken to check her claim. With $\displaystyle \alpha=0.05$ how small would the sample percentage need to be before the claim could legitimately be refuted? (notice that this would invovle a one-tailed test of the hypothesis)

The book mentions that the answer is $\displaystyle \hat{p}<.1342$ but im not sure how they calculated this.

I tried using

$\displaystyle \frac{\hat{p}-.2}{\sqrt{\frac{\hat{p}\cdot (1-\hat{p}}{100}}}=z_{0.05}$ but got really stuck and couldn't calculate it out, so not sure if im on the right track...