Question

assume the random variable ( x ) is normally distributed with mean ( mu = 50 ) and standard deviation ( sigma = 7 ). compute the probability. be sure to draw a normal curve with the area corresponding to the probability shaded.
( p(34lt xlt64) )
which of the following normal curves corresponds to ( p(34lt xlt64) )?
( p(34lt xlt64)=square )
(round to four decimal places as needed.)

Explanation:

Step1: Calculate z - scores

The formula for the z - score is \(z=\frac{x-\mu}{\sigma}\).
For \(x = 34\), \(z_1=\frac{34 - 50}{7}=\frac{-16}{7}\approx - 2.29\).
For \(x = 64\), \(z_2=\frac{64 - 50}{7}=\frac{14}{7}=2\).

Step2: Use the standard normal distribution table

We know that \(P(34\lt X\lt64)=P(-2.29\lt Z\lt2)\).
Since \(P(-2.29\lt Z\lt2)=P(Z\lt2)-P(Z\lt - 2.29)\).
From the standard normal table, \(P(Z\lt2)=0.9772\) and \(P(Z\lt - 2.29)=0.0110\).

Step3: Compute the probability

\(P(-2.29\lt Z\lt2)=0.9772-0.0110 = 0.9662\).

Answer:

\(0.9662\)