Question

assume that z - scores are normally distributed with a mean of 0 and a standard deviation of 1. if ( p(-alt zlt a)=0.4314 ), find a. round to two decimal places

a. 1.49
b. - 0.18
c. 0.33
d. 0.57

Explanation:

Step1: Use symmetry property

Since the standard normal distribution is symmetric about \(z = 0\), \(P(-a<z < a)=2\Phi(a)- 1\), where \(\Phi(a)\) is the cumulative distribution function of the standard normal distribution.
Given \(P(-a < z < a)=0.4314\), then \(2\Phi(a)-1 = 0.4314\).

Step2: Solve for \(\Phi(a)\)

Rearrange the equation \(2\Phi(a)-1 = 0.4314\) to get \(\Phi(a)=\frac{0.4314 + 1}{2}\).

$$ \Phi(a)=\frac{1.4314}{2}=0.7157 $$

Step3: Look up in the standard - normal table

Looking up the \(z\) - value in the standard - normal table (or using a calculator with a normal - distribution function, e.g., in Excel: NORM.S.INV(0.7157)), we find the \(z\) - value corresponding to a cumulative probability of \(0.7157\).

Answer:

C. \(0.33\)