Question

What is the probability of the following
1. P (0< z < 1.46)?​

Answer 1

Approximately
`0.428`.

Explanation:

(This approach requires looking up values on a
`z`-table. Calculators aren't required.)

A typical
`z`-table lists probabilities in the form
`P(Z \le z)` (or equivalently,
`P(Z < z)`,) where
`z\!` is real number (typically greater than or equal to
`0`.)

However, the question asks for a probability in the form
`P(a < Z < b)`. It would be necessary to rewrite this probability for this
`\!z`-table approach to work.

Let
`a` and
`b` denote two real numbers, with
`b > a`.
`P(a < Z < b)` would be equal to
`P(Z < b) - P(Z < a)`. In this question:

`\begin{aligned} & P(0 < Z < 1.46) \\ & = P(Z < 1.46) - P(Z < 0) \end{aligned}`.

Look up the value of
`P(Z < 1.46)` on a
`z`-table:
`P(Z < 1.46) \approx 0.927855`.

On the other hand,
`P(Z < 0) = 0.5` because the Gaussian distribution is symmetric.

Therefore:

`\begin{aligned} & P(0 < Z < 1.46) \\ & = P(Z < 1.46) - P(Z < 0) \\ &\approx 0.927855 - 0.5 \\ &\approx 0.428\end{aligned}`.