SOLUTION: Find the Z-scores that separate the middle 62% of the distribution from the area in the tails of the standard normal distribution.
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-> SOLUTION: Find the Z-scores that separate the middle 62% of the distribution from the area in the tails of the standard normal distribution.
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You can put this solution on YOUR website! let z2 = the high end of the middle distribution.
let z1 = the low end of the middle distribution.
the middle distribution is therefore equal to z2 minus z1.
since the middle distribution is equal to 62%, then you get:
62% = z2 minus z1
the total area of the normal distribution is equal to 100%.
the area outside the middle of the distribution is therefore equal to 100% - 62% = 38%
half of this is in the lower end of the normal distribution and half of this is in the upper end of the normal distribution.
that makes each of those areas equal to 38% / 2 = 19%.
you have 19% of the area under the normal distribution curve to the left of the middle 62% and you have 19% of the area under the normal distribution curve to the right of the middle 62%.
this says that 19% of the area under the normal distribution curve is to the left of z1.
you can find this area using the z-score tables, or you can find it using a z-score calculator.
i found it using a z-score calculator.
the calculator i used was in the TI-84 Plus.
it told me that the z-score that had an area to the left of it equal to .19 was equal to -.8778962906.
since the normal distribution curve is symmetric about the mean, this means that the z-score with an area of .19 to the right of it was .8778962906.
those are the z-scores that separate the middle 62% of the normal distribution curve from the area to the left and to the right of it.
if you look in the z-score table, you will see that the area to the left of a z-score of -.87 is equal to .19214 and the area to the left of a z-score of -.88 is equal to .18943.
the area of .19 is somewhere in between and closer to the z-score of -.88 than it is to the z-score of -.87.
using that calculator, i can simply ask it to find the z-scores that bracket the middle 62% of the normal distribution curve and it tells me the low end z-score and the high end z-score.
here's a display of the use of that calculator to find the answer.
note that the tables give you the area to the left of the z-score.
if you want the area to the right of the z-score, that would be 1 minus the area to the left of the z-score.
for example:
looking at the table, .....
the area of .19 to the left of the z-score is somewhere between z = -.87 and z = -.88.
the area to the right of the z-score of .19 is found by looking for the area of 1 minus .19 = .81 to the left of the z-score.
the z-score is somewhere between z = .87 and z = .88.
you can check this out by looking at the z-score table.
