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X is the weight of a baby when born (in gram). If the distribution of X is N(3315, 575) and Y is the number of babies with weight lower than 2719 in a random sample of 25, then P[Y≤4] is about...?

I'm trying to figure how to solve this. What kind of formula do I use? I suppose N is a normal distribution, that 3315 is the mean and that 575 the variance. Please correct me if I'm wrong.

Thanks!

Edit: As stated in the comments 575 should be the standard deviation. I'm not interested in the answer like someone suggested. If you teach me how should I approach this it is more than enough.

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@Queops, maybe you can edit the question and add in your thoughts and approach so far and where you are getting stuck. – cardinal Feb 7 '11 at 23:35
More likely is that 575 is the standard deviation. You should first ask yourself what would be the probability in a sample of size 1. (Your prof didn't make it a nice even multiple of sd's away from the mean, so I imagine he expected you to use a statistical system or calculator of some sort.) – DWin Feb 7 '11 at 23:36
I suspect that 575 is the population standard deviation, both from the rest of the question and the real Norwegian data in figures 1 and 2 of eb.niehs.nih.gov/bwt/subcfreq.htm – Henry Feb 7 '11 at 23:38
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While we are happy to help with homework questions, this site isn't really aimed at doing your work. Please see the relevant meta question. In particular, how homework questions should be formatted. – csgillespie Feb 8 '11 at 0:01
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@Queops I added tags to provide links to useful threads. As a hint, consider that $Y$ is a random variable that counts a certain number of things out of 25. What distribution do you suppose it might have? What do you need to know about that distribution? The assumptions about the distribution of $X$ are there so you can compute the chance that a single baby has low birth weight. – whuber♦ Feb 8 '11 at 6:09
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1 Answer

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It may be easier to first consider equality for general $Y=y$ and the to "add up" for $y=0,1,2,3,4$.

Because the question as stated does not assume any (logical) dependence between babies in the sample, we can say the all have the same probability of being under the threshold (because we do not know how to differentiate them, given what information there is in the question). Call this probability $P=Pr[X<2719|I]$ (where $I$ denotes the information in the question).

Now this is basically a binomial probability that you are asking for, with $25$ samples, and the probability that you see $y$ exactly is:

$$Pr[Y=y|I]=\frac{25!}{y!(25-y)!}P^y (1-P)^{25-y}$$

And so you simply add this up for your answer:

$$Pr[Y\leq 4|I]=\sum_{y=0}^{4}\frac{25!}{y!(25-y)!}P^y (1-P)^{25-y}$$

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Thanks to everyone. – Queops Feb 8 '11 at 13:16

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