Question

determine p(5 ≤ x ≤ 10) if the mean is 7 and the standard deviation is 1.1.

Explanation:

Step1: Calculate z - scores

The formula for the z - score is \(z=\frac{x-\mu}{\sigma}\), where \(\mu = 7\) (mean) and \(\sigma=1.1\) (standard deviation).
For \(x = 5\): \(z_1=\frac{5 - 7}{1.1}=\frac{-2}{1.1}\approx - 1.82\)
For \(x = 10\): \(z_2=\frac{10 - 7}{1.1}=\frac{3}{1.1}\approx2.73\)

Step2: Use the standard normal distribution table

We want to find \(P(-1.82<Z<2.73)\).
We know that \(P(-1.82<Z<2.73)=P(Z < 2.73)-P(Z<-1.82)\)
From the standard normal distribution table: \(P(Z < 2.73)\approx0.9968\), \(P(Z<-1.82)\approx0.0344\)

Step3: Calculate the probability

\(P(-1.82<Z<2.73)=0.9968 - 0.0344=0.9624\approx0.962\)

Answer:

\(0.962\)