# Null Hypothesis A method in statistics that allow to reject an hypothesis. For example, a school changed the methods they teach and want to evaluate if this change affect the grades of its students. The school generates a histogram of means of student's grades in the past. According to the [[Central limit theorem (CLT)|CLT]] the means generates normal distribution. The Null hypothesis is that the method did not have an affect and so it is expected that the mean of the grades after the change will be inside the 0.95 area of the normal curved. If it is outside it, then we can reject the null-hypothesis and conclude that it is possible that the change had an effect. This method is usable in case that we have collected means of some events (grades of tests) and we have a mean of a new event (grade after some change). Then, we can evaluate how likely the new mean is unexpected w.r.t the original mean normal distribution. ## Binary string example We start with a null hypothesis that event was not generated by a Bernoulli distribution with $p=0.5$. for string with length 25, the mean is in $(0.4,0.6)$ with 0.95 probability. The string $1110110001101111111011110$ with 18 ones has a mean of $18/25=0.72$ and so it is rejected. note that being in the 0.95 area does not promise that the event was sampled from $p=0.5$, for example the string $0101110010010111111001000$ was sampled from Bernoulli distribution with $p=0.4$ and has mean 0f 0.52 that matched the 0.95 area of the normal distribution with $p=0.5$. Another interesting example is the suspicious looking string $0101010101010101010101010$ that has the mean of 0.48 which is in the 0.95 area of the normal distribution with $p=0.5$. i.e., we can't reject it as being sampled from Bernoulli distribution with $p=0.5$ although our suspicions. Reference: [[@What is Random? Chance and Order in Mathematics and Life]] ## Created 2024-05-10 14:31