Question

A binomial experiment is given. Decide whether you can use the normal distribution to approximate the binomial distribution. If you​ can, find the mean and standard deviation. If you​ cannot, explain why. A survey of adults found that 45​% have used a multivitamin in the past 12 months. You randomly select 40 adults and ask them if they have used a multivitamin in the past 12 months.
Select the correct answer below and, if necessary, fill in the answer boxes within your choice.

a) no, because np<5.
b) no because nq<5
c) yes, the mean is ___ and the standard deviation is ___.

Answer 1

Answer:

Yes, The mean is 18 and the standard deviation is approximately 3.146.

Step-by-step explanation:

Consider the provided information.

A survey of adults found that 45​% have used a multivitamin in the past 12 months. You randomly select 40 adults and ask them if they have used a multivitamin in the past 12 months.

n = 40, p = 45% = 0.45, and  q = 1 – 0.45 = 0.55

If np ≥ 5 and np ≥ 5, then the binomial random variable, x is approximately normally distributed, then Mean µ = np and standard deviation σ =$\sqrt{npq}$, where n is the sample size, p is the population proportion, and q = 1 – p.

Now calculate np and nq.

np = (40)(0.45) = 18

nq = (40)(0.55) = 22

Both np and nq are greater than 5, the normal distribution can be used to approximate the binomial distribution.

Thus, µ = np

µ = 40(0.45)=18

Standard deviation, $\sigma= \sqrt{npq}$

$\sigma= \sqrt{(40)(0.45)(0.55)}$

$\sigma= \sqrt{9.9}$

$\sigma=3.146$.

Therefore, the mean is 18 and the standard deviation is approximately 3.146.