# approximating binomial distribution w/ normal distribution

#### fiddlestix327

##### New Member
Hi all,
I'm stuck on the following problem:

The percentage of fat in the bodies of American men is an approximately normal random variable with mean equal to 15% and standard deviation equal to 2%.

a. If these values were used to describe the body fat of men in the United States Army and if 20% or more body fat is characterized as obese, what is the approximate probability that a random sample of 10,000 soldiers will contain fewer than 50 who would actually be characterized as obese?

b. If the army actually were to chec the percentage of body fat for a random sample of 10,000 men and if only 30 contained 20% (or higher) body fat, would you conclude that the army was successful in reducing the percentage of obese men below the percentage of the general population? Explain your reasoning.

I am very stuck. So far, I used the equation mean = np = (10,000)(.15) = 1500. Also, took the square root of npq to get standard deviation of approximately 35.71. However, I don't even know if this is right and I am very much confused by all the figures. Thanks in advance

#### Mean Joe

##### TS Contributor
a. what is the approximate probability that a random sample of 10,000 soldiers will contain fewer than 50 who would actually be characterized as obese?
There's a couple of things going on in this problem. We want to find
P{in a sample of 10,000 soldiers fewer than 50 will be obese}
Define a variable X = # of obese soldiers in a sample of 10,000
So our problem is P{X < 50}.

To get to this, we need to find the P{being obese}.
P{being obese} = P{body fat > 20%}
Define a variable Y = body fat. We are given that Y is normal with mean=15, stdev=2. Thus,
P{being obese} = P{Y > 20%} = P{ z > (20-15)/2 } = P{z > 2.5} = .0062

Taking this, we now calculate mean(of X) = 10,000 * .0062 = 62 and stdev(of X) = sqrt( 10,000 * .0062 * .9938 ) = 7.85

Do you have any questions about what was done so far? From here, you use the normal approximation to the binomial to find P{ X < 50 } ~ P { X <= 49.5 } = ...

#### fiddlestix327

##### New Member
Thank you so much! I understand now =)