I'm not sure what to do here:
A class of 30 students were weighed. Their mean was found to be 58kg with a standard deviation of 5.5kg. What percentage of students will have a mass between 52.5 kg and 63.5 kg?
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I would guess the answer is $\approx 68%$. – Dilip Sarwate Jun 07 '13 at 21:16
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^How did you get to that? I have 3 questions like this, and I have no idea on how to approach it. – missiledragon Jun 07 '13 at 21:26
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1http://img.mit.edu/newsoffice/images/article_images/20120208160239-1.jpg – leonbloy Jun 07 '13 at 21:28
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For the answer, you are expected to assume that the weights have normal distribution. This is a very dubious assumption.
You are also expected to believe that this normal distribution has standard deviation $5.5$.
You are also expected to believe that the probability that such a normal lies within $1$ standard deviation unit of the mean gives us the right percentage. That need not be true. At best it gives an estimate of the percentage that is likely to be not very far from the truth.
But let us hold our noses and make all these assumptions.
From tables of the standard normal, the probability that a normal with mean $\mu$ is $\le \mu+\sigma$, where $\sigma$ is the standard deviation, is approximately $0.8413$. We ust need to look up $\Pr(Z\le 1)$, where $Z$ is standard normal.
So the probability it is between $\mu$ and $\mu+\sigma$ is about $0.2413$. Double this to find the probability of lying between $\mu-\sigma$ and $\mu+\sigma$.
André Nicolas
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Minor typo: that $0.2413$ should be $0.3413$ which gives the $\approx 68%$ that I noted in my comment on the question. – Dilip Sarwate Jun 07 '13 at 22:22