How Do You Calculate y1 and y2 for a Given Probability in a Normal Distribution?

[someguy54]

Need a little help here:

Find the random variable coefficients y1 and y2 where P(y1 < y < y2) = 0.5. Where mean is 0.7 and standard deviation is 0.03 (not sure if you need that). I have no clue where to start with this one.

Thanks for any help

 

[HallsofIvy]

Perhaps because there are an infinite number of answers! -\infty to 0.7 would obviously work because of the symmetry of the normal distribution about the mean. So would 0.7 to \infty. For finite values of y1 and y2, try this. Convert to the "standard" z-score using z= (y- \mu)/\sigma which here is z= (y- 0.7)/0.03. Pick any y1 you want, less than the mean, and calculate its z-score. [For example, choosing (just because it makes the calculation easy) y1 to be 0.67, we get z= -0.03/0.03= -1]. Look that up on a table of the normal distribution (a good one is at http://people.hofstra.edu/Stefan_Waner/RealWorld/normaltable.html ) to find P(y1) [for z= -1 I get 0.46587] If that is less than 0.5, add it to 0.5 to see how much "more" you need and look up the z corresponding to that and, finally, compute the y2 that gives. [0.46587+ 0.5= 0.96587. The table says that corresponds to z= 1.82 and then 1.82= (y2- 0.7)/0.03 gives y2= 0.7546. You can choose any y1 you want, less than 0.7, and do the same to get a different y2.