In a standard normal distribution with mean 0 and standard deviation 1, the value of
for
is approximately
, encompassing 43.14% of the distribution's area between
and
.
Given that
scores are normally distributed with a mean of 0 and a standard deviation of 1, and the probability
= 0.4314 , we'll use the properties of the standard normal distribution to find
.
For a standard normal distribution, the area between
and
under the curve represents the probability
.
Given
, we want to find the value of
.
From the properties of symmetry in the standard normal distribution, the total area between
and
is
, which corresponds to the probability
.
Given:
![\[ P(-a < z < a) = 0.4314 \]](https://img.qammunity.org/2024/formulas/mathematics/high-school/t67e80b77hfepdv8sshfdrxru4e2qp6rkq.png)
Since the total area is
and

![\[ 2a = 0.4314 \]](https://img.qammunity.org/2024/formulas/mathematics/high-school/bvxyuwrmvnvio0nn2gq2htb2o2g00ceedk.png)
![\[ a = (0.4314)/(2) \]](https://img.qammunity.org/2024/formulas/mathematics/high-school/urzynh4x2q5mt1pg16q6krdglmgzvhjssp.png)
![\[ a = 0.2157 \]](https://img.qammunity.org/2024/formulas/mathematics/high-school/88s0zoas1dbjjgep4tqhpf9mdka9yo0qdm.png)
Therefore, the value of
such that
in a standard normal distribution is approximately

complete the question
Assume that z scores are normally distributed with a mean of 0 and a standard deviation of 1. If P (-a < z < a) = 0.4314, find a.