Question

Assume that X is normally distributed with a mean m = $64. If P(X ³ $75) = 0.2981, then the standard deviation of X is approximately equal to

Answer 1

The standard deviation of X is approximately $20.07.

We can solve this problem by using the standard normal distribution and the z-score formula.

First, we need to standardize the variable X by subtracting the mean m and dividing by the standard deviation σ:

z =
`((X - m) )/(σ)`

We want to find the value of σ that makes P(X ≥ $75) = 0.2981. Using the standard normal distribution table, we can find the z-score corresponding to the area to the right of z:

P(Z ≥ z) = 0.2981

z = 0.546

Substituting this value of z and the given mean m, we get:

0.546 = ($75 - $64) /Ф

Solving for σ, we get:

σ = ($75 - $64) / 0.546

σ ≈ $20.07

Therefore, the standard deviation of X is approximately $20.07.