the area between -z and z is equal to .5098.
the z-score in the middle between -z and z is equal to 0.
the area to the left of a z-score of 0 is equal to .5.
the area between -z and 0 is half the area between 0 and z.
because the normal distribution is symmetric, take .5098 and divide it by 2 to get .2549.
the area to the left of 0 is equal to .5
the between -z and 0 is equal to .2549.
subtract .2549 from .5 to get .2451.
that's the area to the left of -z.
solve for -z to get -z = -.6899907576.
that makes z = .699907576.
that's your solution.
the area between -.6899907576 and .699907576 is equal to .5098001472 which is equal to .5098 when rounded to 4 decimal digits.
the solution is confirmed to be good.
visually it looks like this.
2. Find P(-2.11 < Z < -0.37).
the area between -2.11 and -.37 is equal to .3382621848
i think that's what you're looking for.
if it isn't, let me know what it is that you're looking for here.
here's a visual display of what i think you are looking for.
you look up the area to the left of a z-score of -2.11 and you look up the area to the left of a z-score of -.37 and you subtract the smaller area from the larger area to get your anser.
using my calculator, i got the following:
area to the left of a z-score of -2.11 is equal to .0174291159
area to the left of a z-score of -.37 is equal to .3556913007
after between -2.11 and -.37 is equal to .3556913007 minus .0174291159 which is equal to .3382621848.
that's the same as i got before when i asked the calculator for the area between a z-score of -2.11 and -.37 directly.
the calculator i use is the TI-84 Plus to solve problems like these.
here's a visual display.
let meknow if you have any questions or need further clarification on points you don't understand.