Probability in Normal Distribution

I've a question here. Basically I have the mean and standard deviation of a variable (IQ). I want to find the probability that 8 out of 10 people randomly selected have IQ scores between 85.0 and 122.5.

I also have the following information worked out:

• mean = 100
• standard distribution = 15
• n = 10
• p = 0.7745

Should I be using Poisson to do this?

(1) Work out the probability that a single person has an IQ in the range given. You need to make an assumption about the distribution of IQ.

Use the cumulative distribution function for the two end points of the range to calculate the probability of someone's falling between them. Call it $\pi$.

(2) Work out the probability that 8 out of 10 people have an IQ in the range given. You likely want to assume that 'randomly selected' implies independently.

Then you're looking for the probability of an event's occurring in 8 out of 10 independent trials when it has probability $\pi$ of occurring in a single trial. (This phrasing should suggest a suitable distribution.)

• The assumption I made is that it follows a normal distribution. In this case do I still use CDF? – user1275515 Feb 17 '14 at 11:54
• Yes, the normal CDF. – Scortchi Feb 17 '14 at 12:09
• am i right to say that the distribution to be used for the second part is poisson? – user1275515 Feb 17 '14 at 14:08
• (1) Perhaps NIST or IPSUR. (2) The number of events in ten trials can't have a Poisson distribution because you can't get more than ten out of ten whereas a Poisson variable is unbounded above. – Scortchi Feb 17 '14 at 15:03
• You're on the right track: use the binomial distribution. But you don't need to approximate it: use the exact formula. – Scortchi Feb 17 '14 at 18:20