Remember that high alpha level is also associated with high type I error and vise-versa. You may want your type I error to be low when you're dealing with something sensitive. For example, when you're testing whether certain goods have defects or not and you cannot tolerate the defects as the consequences could be fatal. For example, in testing the reliability of the batteries used for pacemakers.
Conversely, high alpha level is ok when you could be relaxed in accpeting the null hypothesis. For example when you insist that there is no difference between the intelligence of north korean and south korean students of the same gender and age groups.. The alpha level should be considered based on personal convictions of how strong you want your evidence to be. The alpha level is the probability or p-value that the researcher is willing to accept as significant. It can also be interpreted as the chance of making a Type 1 or Type 2 error. When you set a more stringent (smaller) alpha level, like .01 or .001, (which decreases the probability of making a Type I error) you increase the likelihood of making a Type II error. Hence, it is suggested that an alpha level of .05 is a good compromise between the likelihoods of making Type I and Type II errors.
Low Alpha = Low probability of a Type I error, which means that the consequences of rejecting a true null hypothesis could be dire. Classic example (though not an experiment) is a jury trial. Null is innocence. Rejecting that null is