μ = 15.8; σ = 3.8 P(10 ≤ x ≤ 26)

Question

Assume that x has a normal distribution with the specified mean and standard deviation. Find the indicated probability.

Answer 1

Recall that z-score of X is Z=(X-mu)/sigma.

Step 1: Compute the z-scores of 10 and 26.

That is: the z-score of 10 is (10-15.8)/3.8=-5.8/3.8=-1.53 (round to two decimals)

and the z-score of 26 is (26-15.8)/3.8=10.2/3.8=2.68 (round to two decimals).

Step 2: So P(10<X<26)=P(-1.53<Z<2.68)=P(Z<2.68)-P(Z<-1.53) =P(Z<2.68)-P(Z>1.53).

Using Z-table, we have P(Z<2.68)=0.9963 and P(Z>1.53)=0.0630. So the answer is 0.9963-0.0630=0.9333.