I'm a bit confused when it comes to setting up the null and alternate hypotheses. Should we try to support the value of the sample mean when setting up the alternate hypothesis, or verify the claim that is given in the question?
E.g. The manager would like to use automatic cashiers, however, she believes that at least 50% of customers prefer human cashiers. In the sample of 75 customers, 28 said they prefer automatic cashiers. Is there sufficient evidence that the manager's belief is wrong?
p̂ = 47/75 = 0.63
p0 = 0.50
If we try to prove that the sample mean is true, then the null and alternate hypothesis would be:
Ho: p0<=0.5
Ha: p0>0.5
If we use the claim in the question, then null and alternate hypothesis would be:
Ho: p0>=0.5
Ha: p0<0.5
Which one is correct and why?
Usually the alternative hypothesis is what we wish to show. In this case we wish to show that the manager's claim is incorrect. Letting $p$ denote the proportion of customers who prefer human cashiers, the manager being incorrect would mean $p<0.5$
We have
$$H_0: p\geq0.5$$
$$H_a: p\lt0.5$$
Since $\hat{p}=\frac{47}{75}\approx0.627$ we certainly don't have evidence to reject the null hypothesis.
The more interesting question would be to test whether the manager's claim is correct. However, the wording of the problem indicates to me that these are the hypotheses.
Note: You cannot accept a null hypothesis. You can only fail to reject it or reject it.