SOLUTION: If a normal distribution with mean and variance both more than zero has 46th percentile equal to 20 times multiple of standard deviation, then what is mean in terms of standard d

Question 1150739: If a normal distribution with mean and variance both more than zero has 46th
percentile equal to 20 times multiple of standard deviation, then what is mean
in terms of standard deviation?
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
formula to use is z = (x - m) / s
x is the z-score
x is the raw score.
m is the raw mean.
s is the standard deviation.

both the mean and the variance are greater than 0.
since the standard deviation is the square root of the variance, then the standard deviation is also greater than 0.

p = the percentile.
when p = .46, then 46% of the area under the normal distribution curve is to the left of the z-score.
when p = .46, you can use your z-score calculator to find that z = -.1004337118.

at the 46th percentile, z = -.1004337118.

the formula of z = (x - m) / s becomes -.1004337118 = (x - m) / s at the 46th percentile.

multiply both sides of this formula by s to get:

-.1004337118 * s = x - m

solve for x to get x = -.1004337118 * s + m

at the 46th percentile, x = -.1004337118 * s + m and also x = 20 * s.

formula of x = -.1004337118 * s + m becomes 20 * s = -.1004337118 * s + m

solve for m to get m = 20 * s + .1004337118 * s = 20.1004337118 * s. *****

at the 46th percentile, you get m = 20.1004337118 * s.

when s = 1, m = 20.1004337118

when s = 50, m = 50 * 20.1004337118 = 1005.0216855898

at the 46th percentile, your z-score will be z = -.1004337118

when s = 1, the formula z = (x - m) / s becomes -.1004337118 = (x - 20.1004337118) / 1.
solve for x to get x = -.1004337118 * 1 + 20.1004337118 = 20.

when s = 50, the formula z = (x - m) / s becomes -.1004337118 = (x - 1005.0116855898) / 50.
solve for x to get x = -.1004337118 * 50 + 1005.0116855898 = 1000.

when x = 20 and s = 1, then x = 20 * s.

when x = 1000 and s = 50, then x = 20 * s as well.

the value of x at the 46th percentile will always be 20 * s.

the formula holds up and the mean in terms of s will always be m = 20.1004337118 * s.

that's your solution as far as i can tell.

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