## Problem 1188

A binomial experiment is given. Decide whether you can the normal distribution to approximate the binomial distribution. If you can, find the mean and standard deviation. If you cannot, explain why.

A  survey a adult found that 58% have used a multivitamin in the past 12 months. You randomly select 55 adults and ask them if they have used a multivitamin in the past 12 months.

Solution:-

If np ≥ 5 and np ≥ 5, then the binomial random variable, x is approximately normally distributed, with mean µ = np and standard deviation σ = , where n is the sample size, p is the population proportion, and q = 1 – p.

Find np and nq to determine if the normal distribution can be used to approximate the binomial distribution. First determine the values of  n, p, and q.

n = 55

p = = 0.58

q = 1 – 0.58 = 0.42

Now calculate np and nq.

np = (55)(0.58) = 31.9

nq = (55)(0.42) = 23.1

Since both np and nq are greater than 5, the normal distribution can be used to approximate the binomial distribution. Thus, calculate the mean, µ = np, and standard deviation, σ = . Recall that up = 31.9. Find the standard deviation, rounding to two decimal places.

σ =

Therefore, the mean is 31.9 and the standard deviation is approximately 3.66.