Question

Probability and Statistics: Gaussian Distribution: Suppose that a certain sample space of numbers can be modeled by a Gaussian distribution with a mean of 68 and a standard deviation of 2.5. First, find the percentage of the sample space whose numbers is between 66 and 71. Second, find the percentage whose numbers is approximately 72.

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For the first one, I tried to normalize 68 into -0.8 and 71 to 1.2 (using (parameter-mean)/(std. dev)), then I tried to numerically integrate the standard Gaussian PDF from -0.8 to 1.2 to get 67.31%. Is this right?

Answer 1

I'm not sure how to do the second one, though..

Answer 2

I get 67.30% for the first one using a standard normal distrubution table.
I will post a method and a solution for the second one. Please wait a few minutes.

Answer 3

Thank you very much.

Answer 4

The percentage of numbers approximately equal to 72 can be found by appropriate calculation of the z-scores for X above and below 72.
For example if the numbers lie within 72 plus and minus 0.5:
\[z _{1}=\frac{71.5-68}{2.5}\]
\[z _{2}=\frac{72.5-68}{2.5}\]
Using these values of z-score the required percentage = 4.49%

Answer 5

How did you decide to use 0.5 though? Is this number arbitrary?

Answer 6

Good question! 0.5 is a correction used when changing from a discrete to a continuous distribution such as in the normal approximation of the binomial distribution. Although it is not directly relevant to your question, I thought it would be a good starting point. You can try tightening the limits by using smaller values than 0.5 and assessing the senstitivity if you like.

Answer 7

This is really late, but thank you very much for your help!

Answer 8

You're welcome :)