I'm following a statistics course for the first time and wondering how can I calculate the p-value from the information given in the picture. I read the explanation but still couldn't understand how it works (I'm wondering how to do it by hand because I have a test tomorrow and by hand is the way to go).
On
they have used the standard normal distribution to calculate $p(z>1.66)$ which you can obtain from tables or from a calculator. The p-value is $0.048457....$ which is less than $5$%
Note the use of the $>$ symbol in this case. This is because we are trying to determine if the $33$ is in the upper critical region, since $33$ is bigger than the mean which is $300\times0.08=24$
Note: $$H_0: p=0.08; H_a: p>0.08$$ The null hypothesis can be rejected if $z>z_{\alpha}$ or $p<\alpha$. Hence: $$z=1.66>1.645=z_{0.05} \Rightarrow \text{Reject} \ H_0.$$ $$p=P(z>1.66)=0.048<0.05=\alpha \Rightarrow \text{Reject} \ H_0.$$ To find $p$ value or $z_{\alpha}$ usually a standard normal table is provided on the statistics tests, because it is not feasible to find them manually (except for some popular ones).