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Null hypothesis:

$H_0:p=0.8$

This means that the genuine proportion of all consumers enrolling in a shop loyalty card program is less than 0.8.

Null hypothesis:

$H_0:p>0.8$

This means that the genuine proportion of all consumers enrolling in a shop loyalty card program is more than 0.8.

Rejection region for $伪=0.01$

The test is right-tailed, and the level of significance is $伪=0.01$

The critical value for the right-tailed test $伪=0.01$is $z_{0.01}=2.33$. Hence, the region is $z_c>2.33$

Therefore,

$$\begin{gathered} z=\frac{\hat{p}-p}{{\displaystyle \frac{\sqrt{0.8(1-0.8)}}{250}}} \\ 0.0589=\hat{p}-0.8 \\ \hat{p}=0.8589 \end{gathered}$$

Thus, the rejection region is $\hat{p}>0.8589$

$p=79\%$

Power =$P(\overline{p}>0.858\overline{)9}p=0.79)$

$$\begin{gathered} =P\left(z>\frac{0.8589-0.79}{\sqrt{{\displaystyle \frac{0.79(1-0.79)}{250}}}}\right) \\ =P\left(z>\frac{0.0689}{0.0258}\right) \\ =P(z>2.67) \\ =0.5-0.4962 \\ =0.0038 \end{gathered}$$

Thus, if the genuine proportion of consumers enrolling in a loyalty card program is 79%, the probability that the test findings will back up the claim is 0.0038.