Question

If `X` is normally distributed, how do I find `"P"(X<5.83)`?

Answer 1

Find the corresponding `z` point on the `Z`-distribution that matches up with `x=5.83` from the `X`-distribution (if necessary). Look up this value in a `z`-table.

Explanation:

If the question is asking you to use the standard normal distribution `Z` (centered at `mu = 0` and with a variance of `sigma^2=1`), then the answer can be calculated in two ways:

1. Look up 5.83 in a `z`-table. Unfortunately, no `z`-table in wide publication will list values higher than 3.5, because at that point, the value they give is within 0.01% of 100%. Higher `z`-values than 3.5 are hardly practical.

2. Use computer software (or a web utility) to find out what percentage of the standard normal distribution is to the left of `x=5.83`. Using R, I found that the area under the curve to the left of `x=5.83` is approximately 0.99999999722863. In other words, ridiculously close to 1.

This leads me to think that perhaps the question is using a non-standard normal distribution. If that is the case, you can still calculate `P(X<5.83)` as above, but we first need to find the corresponding point on the standard normal distribution. This is done using the following formula:

`z=(x-mu)/sigma`

Subtracting `mu` shifts a non-standard normal distribution so that it is centered at 0; dividing this difference by `sigma` stretches the distribution out (like a rubber band) from 0 on both sides, so that its shape matches that of the standard normal distribution. Thus, for example, if `X~"Norm"(mu=3, sigma^2=4)`, then

`P(X<5.83)=P(Z<(5.83-mu)/sigma)`
`color(white)(P(X<5.83))=P(Z<(5.83-3)/2)`
`color(white)(P(X<5.83))=P(Z<2.83/2)`
`color(white)(P(X<5.83))=P(Z<1.415)`.

This `P(Z<1.415)` is then found by looking up the area to the left of `Z=1.415` in a `z`-table.