When we reject and accept the null Hypothesis?
Your values for Z are correct. The null hypothesis is rejected if the value for Z is “more extreme” (meaning it has greater absolute value) than the “critical value”, which is the Z-value for the significance level. The significance level is usually set at 0.05 (5%) but is sometimes set greater or less, and it is the frequency in which the observed data is expected from a random model. Both your examples are “one-sided” (or “one-tailed”) tests, where the alternate hypothesis is that the mean of the population is strictly less than what is stated in the null hypotheses, not just different from it. The critical value Z(0.05) for a one-sided 0.05 significance level is ±1.645, because the probability of a normally-distributed random variable (“Z”) being farther from zero than 1.645 units is 0.05, which your first Z-value -1.111 is not (so you don’t reject) but the second Z-value -6 is (so you reject).
