Question

If \(X\) is a normal random variable with mean \(\mu=100\) and variance \(\sigma^2=25\), then \[ P(90<X<120) \] is same as ____.

• A. \(P(-1<Z<1)\)
• B. \(P(-2<Z<4)\)
• C. \(P(4<Z<4.1)\)
• D. \(P(-2<Z<3)\)

Hint:

To standardize a normal random variable, \[ \boxed{ Z=\frac{X-\mu}{\sigma} } \] where \[ \mu=\text{Mean},\qquad \sigma=\text{Standard deviation}. \]

Answer 1

The Correct Option is B

Concept: For a normal random variable, \[ \boxed{ Z=\frac{X-\mu}{\sigma} } \] converts the variable into the standard normal distribution.

Step 1: Compute the standard deviation. Given, \[ \sigma=\sqrt{25}=5. \]

Step 2: Convert the limits into \(Z\)-scores. For \(X=90\), \[ Z=\frac{90-100}{5} =-2. \] For \(X=120\), \[ Z=\frac{120-100}{5} =4. \] Hence, \[ P(90<X<120) = P(-2<Z<4). \] Therefore, \[ \boxed{P(-2<Z<4).} \] Thus, the correct option is \[ \boxed{(B)\;P(-2<Z<4).} \]