The percent of fat calories that a person in America consumes each day is normally distributed with a mean of about 36 and a sta
ndard deviation of about 10. Suppose that 16 individuals are randomly chosen. Let x represent the average percent of calories in the sample. For the group 16, find the probability that the average percent of fat calories consumed is more than 37
Answer: Normal approximation can be used for discrete sampling distributions, such as Binomial distribution and Poisson distribution if certain conditions are met.
Step-by-step explanation: We will give conditions under which the Binomial and Poisson distribitions, which are discrete, can be approximated by the Normal distribution. This procedure is called normal approximation.
1. Binomial distribution: Let the sampling distribution be the binomial distribution , where is the number of trials and is the probability of success. It can be approximated by the Normal distribution with the mean of and the variance of , denoted by if the following condition is met:
2. Poisson distribution: Let the sampling distribution be the Poisson distribution where is its mean. It can be approximated by the Normal distribution with the mean and the variance , denoted by when is large enough, say (however, different sources may give different lower value for but the greater it is, the better the approximation).