Assume that $z$-scores are normally distributed with a mean of 0 and a standard deviation of 1. If $P(-b < z < b) = 0.9544$, then what is the value of $b$?
$b=\square$
Answer (2)
To find b such that P ( − b < z < b ) = 0.9544 , we set up the equation and solve for b , leading to b ≈ 2.05 .
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Use the symmetry of the normal distribution to express P ( − b < z < b ) as 2 P ( z < b ) − 1 .
Set up the equation 2 P ( z < b ) − 1 = 0.2676 .
Solve for P ( z < b ) to get P ( z < b ) = 0.6338 .
Find the corresponding z -score b such that P ( z < b ) = 0.6338 , which is approximately 0.3419 .
Explanation
Understand the problem and provided data We are given that z -scores are normally distributed with a mean of 0 and a standard deviation of 1. We are also given that $P(-b
