Question

23 fill in the blank 1 point assume the random variable x is normally distributed with mean μ = 83 and standard deviation σ = 5. find the indicated probability. p(68 < x < 78) = 0.1574 (round to four decimal places as needed.)

Explanation:

Step1: Calculate the z - scores

The formula for the z - score is \(z=\frac{x-\mu}{\sigma}\).
For \(x = 68\), \(z_1=\frac{68 - 83}{5}=\frac{-15}{5}=-3\).
For \(x = 78\), \(z_2=\frac{78 - 83}{5}=\frac{-5}{5}=-1\).

Step2: Use the standard normal distribution table

We know that \(P(68\lt x\lt78)=P(-3\lt z\lt - 1)\).
Since \(P(-3\lt z\lt - 1)=P(z\lt - 1)-P(z\lt - 3)\).
From the standard normal distribution table, \(P(z\lt - 1)=0.1587\) and \(P(z\lt - 3)=0.0013\).

Step3: Calculate the probability

\(P(-3\lt z\lt - 1)=0.1587 - 0.0013=0.1574\).

Answer:

\(0.1574\)