standard normal curve

Question

Assume that z-scores are normally distributed with a mean of 0 and a standard deviation of 1.

If P(−b<z<b)=0.967P(-b<z<b)=0.967, find b.

Answer 1

Note that a normal standard curve is symmetric on both sides. The probability corresponds to the area under the curve.

Due to symmetry on both sides (-b < 0 < b) [-b......0......b], we can solve

P(0 < Z < b) = 0.967*(0.5) = 0.4835.

Note that P(mean) = P(Z > 0) = 0.5 = P (Z < 0)

P(0 < Z < b) = P(Z < b) - P(Z < 0) = 0.4835 --> P(Z < b) = 0.4835 + 0.5 = 0.9835.

Using Z table (input mean = 0, standard deviation = 1, and cumulative probability = 0.,9835), we find that b = 2.135 and -b = -2.135....

Therefore, P (-2.135 < Z < 2.135) = 0.967. This means that the probability a range of values deviates from the mean in both directions by 2.135 standard deviations is 96.7%.