Question

question 10
find the area between ( x = 14.48 ) and ( x = 16.54 ) under the normal distribution curve with ( mu = 12 ) and ( sigma = 2 ).

Explanation:

Step1: Calculate z - scores

The formula for the z - score is \(z=\frac{x-\mu}{\sigma}\).
For \(x = 14.48\), \(z_1=\frac{14.48 - 12}{2}=\frac{2.48}{2}=1.24\).
For \(x = 16.54\), \(z_2=\frac{16.54 - 12}{2}=\frac{4.54}{2}=2.27\).

Step2: Use the standard normal distribution table

We know that the area under the standard normal curve \(P(Z<z)\) can be found from the standard - normal table.
\(P(1.24<Z<2.27)=P(Z < 2.27)-P(Z < 1.24)\)
From the standard - normal table, \(P(Z < 2.27)=0.9884\) and \(P(Z < 1.24)=0.8925\)

Step3: Calculate the difference

\(P(1.24<Z<2.27)=0.9884 - 0.8925=0.0959\)

Answer:

\(0.0959\)