Question

It is given that \(\text{Prob}(-1 \leq z \leq 1)=0.683\), \(\text{Prob}(-2 \leq z \leq 2)=0.954\), and \(\text{Prob}(-3 \leq z \leq 3)=0.997\), when \(z\) follows a standard normal distribution. If \(X\) follows a normal distribution with mean and variance as \(5\) and \(4\), respectively, then \(\text{Prob}(1 \leq X \leq 7)=\underline{}\) (rounded off to three decimal places).

Hint:

For a normal distribution, first convert \(X\) into \(z\) using \(z=\frac{X-\mu}{\sigma}\), then use standard normal probabilities and symmetry.

Answer 1

Correct Answer: 0.819

Step 1: Note the spread of X.
Here $X$ has mean $5$ and variance $4$, so the standard deviation is $\sigma=2$.

Step 2: Turn the limits into z scores.
Using $z=\dfrac{X-\mu}{\sigma}$, the value $X=1$ gives $z=\dfrac{1-5}{2}=-2$, and $X=7$ gives $z=\dfrac{7-5}{2}=1$. So we want $\text{Prob}(-2\le z\le1)$.

Step 3: Split the range at zero.
Write it as $\text{Prob}(-2\le z\le0)+\text{Prob}(0\le z\le1)$.

Step 4: Use symmetry on the given numbers.
From $\text{Prob}(-2\le z\le2)=0.954$, half of it is $0.477$. From $\text{Prob}(-1\le z\le1)=0.683$, half is $0.3415$.

Step 5: Add the two pieces.
\[ 0.477+0.3415=0.8185\approx0.819 \]
\[ \boxed{0.819} \]