statistics

The margin of error E for this group of scientists would be calculated as: E = 1.96 x √(52/150) = 0.503. This means that if the probability of success of the electric shock treatment to cure paranoid delusions is p, then the confidence interval for this treatment is calculated to be (p - 0.503, p + 0.503). The critical value of 1.96 for a 95% confidence level comes from the standard normal distribution. This distribution gives us the probability that a result is true based on the probability of the result occurring along a normal distribution curve. The confidence level of 95% means that there is a 5% chance that the result could be outside of the confidence interval. Therefore, using the critical value of 1.96 helps to ensure that there is less than a 5% chance that the true probability of success is outside of the confidence interval.