8.1 Normal Approximations Chapter 8 8.1 Normal Approximations

Normal Approximations As the number of trials n gets larger, the binomial distributions gets close to a normal distribution. At a certain point, it becomes difficult to calculate binomial probabilities on the calculator. Therefore, it is more advantageous to use Statistical Software or make an approximation using a normal distribution.

The Normal Approximation Formula N(µ, ) = N (np, ) What will determine when to use Normal approximation as opposed to the Binomial distribution formulas? np ≥ 10 and n(1 - p) ≥ 10.

Example: Page 455: 8.19 19 A) X = 400; Yes b/c each response falls into a category of approve/disapprove, there are a fixed number of n observations, these observations are independent, and each probability is equally likely. B) binomcdf(400, .92, 358) = .0441 C) mean = (400*.92) = 368 standard deviation = squ rt (368*.08) = 5.426 D) N(368, 5.426) = The difference between the two answers is 0.0112. The approximation is ….

Binomial Simulations We can calculate the probabilities of a binomial event if the random variable X and “success” are defined, the probability of success is given and we know the number of trials. Therefore, we only conduct simulations to help convince the reader, who may know no statistics, of what is occurring in the problem.

Binomial Simulations Since binomial distributions rely on success and failure we can use the command RANDBIN (math:prb:7), to help generate results. Assign success to either 0 or 1 but normally a 1 is used.

Assignment Exercises 8.17, 8.20, and 8.22