Question

the area under the normal curve between ( z = 0 ) and ( z = 1 ) is ______ the area under the normal curve between ( z = 1 ) and ( z = 2 ).
less than
greater than
equal to
less than, greater than, or equal to, depending on the value of the mean,
less than, greater than, or equal to, depending on the value of the standard deviation

Explanation:

Step1: Recall the properties of the normal distribution

The normal distribution is symmetric about \(z = 0\). The area under the normal curve is given by the cumulative - distribution function \(P(a<Z < b)=\varPhi(b)-\varPhi(a)\), where \(\varPhi(z)\) is the cumulative distribution function of the standard normal distribution \(Z\sim N(0,1)\).
We know that \(\varPhi(z)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{z}e^{-\frac{t^{2}}{2}}dt\).

Step2: Calculate the areas

The area between \(z = 0\) and \(z = 1\) is \(P(0<Z<1)=\varPhi(1)-\varPhi(0)\).
From the standard - normal table, \(\varPhi(0) = 0.5\) and \(\varPhi(1)=0.8413\), so \(P(0 < Z < 1)=0.8413 - 0.5=0.3413\).
The area between \(z = 1\) and \(z = 2\) is \(P(1<Z<2)=\varPhi(2)-\varPhi(1)\).
From the standard - normal table, \(\varPhi(2)=0.9772\) and \(\varPhi(1) = 0.8413\), so \(P(1 < Z < 2)=0.9772-0.8413 = 0.1359\).

Answer:

greater than