Question

assume the random variable x is normally distributed with mean μ = 50 and standard deviation σ = 7. compute the probability. be sure to draw a normal curve with the area corresponding to the probability shaded.
p(34 < x < 57)
which of the following normal curves corresponds to p(34 < x < 57)?
p(34 < x < 57) = □
(round to four decimal places as needed.)

Explanation:

Step1: Standardize the values

For \(x = 34\), the \(z\)-score is \(z_1=\frac{34 - 50}{7}=\frac{-16}{7}\approx - 2.29\).
For \(x = 57\), the \(z\)-score is \(z_2=\frac{57 - 50}{7}=\frac{7}{7}=1\).

Step2: Use the standard normal distribution table

We know that \(P(34\lt X\lt57)=P(-2.29\lt Z\lt1)\).
By 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).
From the standard normal table, \(\Phi(1)=0.8413\) and \(\Phi(-2.29) = 0.0110\).

Step3: Calculate the probability

\(P(-2.29\lt Z\lt1)=\Phi(1)-\Phi(-2.29)=0.8413 - 0.0110=0.8303\).

Answer:

\(0.8303\)