# Self-Study question on Normal Distribution

I was attempting a self-study question on the normal distribution, but I wasn't able to get to the answer. I'm not sure if my method used here is incorrect, and would appreciate some guidance and advice.

Question

The pulse rate per minute of the adult male population between 18 and 25 years of age in the United States is known to have a normal distribution with a mean of 72 beats per minute and a standard deviation of 9.62. The requirements for military service state that anyone with a pulse rate over 100 is medically unsuitable for service. In a group of 20 males between 18 and 25 years of age, what is the probability that at least one of them would be declared unfit because their pulse rates are too high? Assume that their pulse rates are independent.

My approach was as follows:

$$X \sim \mathcal{N}(72, 9.62^2)$$

With $$n = 20$$, find $$\Pr(X>100)$$. Since the value is normally distributed, I choose to normalize the information and use $$Z$$ to find my value.

$$\Pr[Z > (100-72)/9.62] = \Pr(Z > 2.9106)$$. Using the $$Z$$-table, I deduced that the value is 0.00181.

This answer, though, is far off from in the guide's answer key, which is 0.0356. The answer key does not work through the steps, so there is no information about where I went wrong.

However, this answer seemed to be wrong based on the study guide. Appreciate some guidance and directions please.

• Sometimes you can get a hint by comparing your answer to the "official" one. For instance, when the answers differ by a factor of two, that's usually because one of you is performing a two-sided test and the other is performing a one-sided test. That's not the case here, though. In your situation $0.0356/0.00181 = 19.7$ is suggestively close to $20$, the number of males in the group. That is a useful clue. – whuber Apr 21 '14 at 21:26
• Thanks whuber. I attempted to multiply it by 20, but still no avail. Should I attempt Hypergeometric Distribution? Slightly confused here. Many thanks in advance for the guidance please. – user1275515 Apr 22 '14 at 5:37

## 1 Answer

What you did was correct but you have only completed step 1 of 2. The question asks of the the 20 males what is the probability at least one will be unfit. You have not answered this part.

You have calculated the probability any given male is unfit. Assuming the males are independent and the probability stays constant, you now have 20 trials and you need to find P(X>0) using a discrete distribution.

• Thanks Glen. Appreciate your kind help. What I attempted was to then take 20(0.00181), but I was still not able to get to the answer. I was wondering if you will be able to advice on what I am getting wrong? Is this something that I should use Hypergeometric Distribution instead? – user1275515 Apr 22 '14 at 5:35
• Use the binomial distribution with n=20 and p=.00181. Then find P(X>0) or equally 1-P(X=0). – Glen Apr 22 '14 at 15:34