Each visit to amazon.com has a probability p of resulting in a purchase. Out of a random sample of 500 visits, 15 results in purchases. Is this statistically significant evidence that p ≠ 2%? Describe relevant hypotheses and statistical test.
I think the starting point should be to define null and alternative hypothesis. So, null hypothesis will be p ≠ 2% and the alternative p = 2%. But I don't know how to continue.
Thanks in advance.
The two hypotheses are: $H_0: p=0.02$ and $H_1: p \neq 0.02$
The approximated two-sided confidence interval is
$$\large{\left[\hat p-z_{(1-\frac{\alpha}{2})}\cdot \sqrt{\frac{\hat p\cdot (1-\hat p)}{ n}} ; \ \hat p+z_{(1-\frac{\alpha}{2})}\cdot \sqrt{\frac{\hat p\cdot (1-\hat p)}{ n}}\right]}$$
(Here we apply the central limit theorem to approximate the interval).
where $\hat p=\frac{15}{500}=0.03, n=500$
Often the confidence level of 95% is used (Other values can be used as well. It depends on the situation). In this case $\alpha=0.05$. Thus we hat to look up in a table of a standard normal distribution what $z_{1-0.025}=z_{0.975}$ is. It is $1.96$
If the value 0.02 is inside the interval we do not reject the nullhypothesis. I think you can proceed from here.