Question

Find the value of x so that the area under the normal curve between µ and x is 0.4525 and the value of x is less than µ.

Answer 1

Let x be a continuous random variable that follows a normal distribution with a mean of and a standard deviation of 25. Find the value of x so that the area under the normal curve between µ and x is 0.4525 and the value of x is less than µ.

$P(x< X < \mu) = 0.4525 \\ P(x < X < 200) = 0.4525 \\ P(X<200) -P(X<x) = 0.4525 \\ P(Z< \frac{200-200}{25}) -P(Z< \frac{x-200}{25}) = 0.4525 \\ P(Z< 0) -P(Z< \frac{x-200}{25}) = 0.4525 \\ 0.5 -0.4525 = P(Z< \frac{x-200}{25}) \\ P(Z< \frac{x-200}{25}) = 0.0475 \\ \frac{x-200}{25} = -1.6696 \\ x -200 = 25 \times (-1.6696) \\ x = 200 - 41.74 \\ x = 158.26$