Question

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

Explanation:

Step1: Standardize the values

To standardize \(x = 68\) and \(x = 78\) using the formula \(z=\frac{x-\mu}{\sigma}\).
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

We know that \(P(68\lt x\lt78)=P(-3\lt z\lt - 1)\)
Using the property \(P(a\lt z\lt b)=\Phi(b)-\Phi(a)\) (where \(\Phi(z)\) is the cumulative distribution function of the standard - normal distribution)
\(P(-3\lt z\lt - 1)=\Phi(-1)-\Phi(-3)\)
From the standard - normal table, \(\Phi(-1)=0.1587\) and \(\Phi(-3)=0.0013\)

Step3: Calculate the probability

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

Answer:

\(0.1574\)